process control department of chemical and materials

Economic Model Predictive Control with Extended Horizon

by

Su Liu

A thesis submitted in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

in

Process Control

Department of Chemical and Materials Engineering

University of Alberta

c©Su Liu, 2017

Abstract

In this thesis, we propose a computationally efficient economic model predictive con-

trol (EMPC) design which is based on a well-known methodology — the separation of

the control and prediction horizon. The extension of the prediction horizon of EMPC

is realized by employing an auxiliary control law which asymptotically stabilizes the

optimal steady state. The contributions of this thesis are to systematically analyze

the stability and performance of the general EMPC scheme with extended horizon,

and to explore its extensions and applications to several specific scenarios.

Specifically, we establish stabilizing conditions of the proposed EMPC in a pro-

gressive manner. First, we establish practical stability of the EMPC with respect

to the extended horizon for strictly dissipative systems satisfying mild assumptions.

Then, under stronger conditions involving Lipschitz continuities and exponential sta-

bility of the auxiliary controller, the shrinkage of the practical stability region is

shown to be exponential. Further, we characterize a general condition on the stor-

age function under which exponential stability of the optimal steady state can be

established. Conventional set-point tracking MPC with quadratic cost falls into the

latter category. The achievable performance of the proposed EMPC design is also

discussed in a similar manner under different stabilizing conditions. It is revealed

that the performance of the proposed EMPC is approximately upper-bounded by the

auxiliary controller. Our theoretical results provide valuable insights into the intrinsic

properties of EMPC as the discussions are laid out in a very general setting and the

results are compatible with the analysis of existing MPC / EMPC designs.

With a deepened theoretical understanding of EMPC with extended horizon, we

further explored the extension of the proposed EMPC design in several scenarios: (i)

We consider the case where a locally optimal LQR control law can be found. A termi-

nal cost is constructed as the value function of the LQR controller plus a linear term

ii

characterized by the Lagrange multiplier associated with the steady-state constraint.

This design results in an EMPC this is locally optimal. (ii) We take advantage of

EMPC with extended horizon to handle systems with scheduled switching operations.

The EMPC operations are divided into two phases — an infinite-time operation phase

and a mode transition phase. The proposed EMPC schemes are much more efficient

and achieves improved mode transition performance than existing EMPC designs.

(iii) We consider systems with zone tracking objectives which can be viewed as a

special case of economic objective. The proposed zone MPC penalizes the distance

of the predicted state and input trajectories to a desired target zone which is not

necessarily positive invariant. We resort to LaSalle’s invariance principle and develop

invariance-like theorem which is suitable for stability analysis of zone control. Auxil-

iary controllers which asymptotically stabilize an invariant subset of the target zone

are employed to enlarge the region of attraction. (iv) We apply the proposed EMPC

algorithm to the control of oilsand primary separation vessel (PSV) to maximize the

bitumen recovery rate.

iii

Preface

The materials presented in this thesis are part of the research project under the

supervision of Dr. Jinfeng Liu, and is funded by Alberta Innovates Technology Futures

(AITF) and Natural Sciences and Engineering Research Council (NSERC) of Canada.

Chapter 2 of this thesis is a revised version of Su Liu and Jinfeng Liu, Economic

model predictive control with extended horizon. Automatica, 73:180-192, 2016.

Chapter 3 of this thesis is a revised version of Su Liu and Jinfeng Liu. A terminal

cost for economic model predictive control with local optimality. In Proceedings of

the American Control Conference, pages 1954-1959, Seattle, WA, 2017.

Chapter 4 of this thesis is a revised version of Su Liu and Jinfeng Liu. Economic

model predictive control for scheduled switching operations. In Proceedings of the

American Control Conference, pages 1784-1789, Boston, MA, 2016.

Chapter 5 of this thesis is a revised version of Su Liu, Yawen Mao, and Jinfeng

Liu, Nonlinear model predictive control for zone tracking. IEEE Transactions on

Automatic Control, submitted.

Chapter 6 of this thesis is a revised version of Su Liu, Jing Zhang and Jinfeng Liu.

Economic MPC with terminal cost and application to an oilsand primary separation

vessel. Chemical Engineering Science, 136:27-37, 2015.

iv

Acknowledgements

Foremost, I would like to express my sincere gratitude to my supervisor Professor

Jinfeng Liu for providing me with an excellent atmosphere for doing research. He

has been a role model to me as a researcher, mentor and instructor. This thesis

would not have been possible without his competent guidance, constant availability

and wholehearted support.

Besides my supervisor, I would like to thank my committee members Profes-

sor Vinay Prasad, Professor Biao Huang, Professor Qing Zhao and Professor Luis

Ricardez-Sandoval for their interest in my work.

I would also like to thank members from our research group who have helped me

along the way: Jing Zhang, Xunyuan Yin, Tianrui An, Kevin Arulmaran, Jayson

McAllister, An Zhang, Nirwair Bajwa, Yawen Mao, Jannatun Nahar and Benjamin

Decardi-Nelson. I also wish to thank my friends who have made life at the University

of Alberta a wonderful experience: Lin Shao, Lei Yao, Ruomu Tan, Wenhan Shen,

Shunyi Zhao, Tianbo Liu, Xin Xu.

I gratefully acknowledge the financial support from Natural Sciences and Engi-

neering Research Council of Canada (NSERC) and Alberta Innovative Technology

Futures (AITF).

I would like to thank my girlfriend, Huan Huang, for her companion on our journey

of self-discovery. Last but not least, my deepest gratitude goes to my mother, Yuehe

Zhou, whose unconditional love and inner beauty has been my greatest source of

strength.

v

Contents

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Research overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 A brief review of MPC . . . . . . . . . . . . . . . . . . . . . . 2

1.2.2 Separation of control and prediction horizon . . . . . . . . . . 4

1.2.3 Economic MPC . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Contributions and thesis outline . . . . . . . . . . . . . . . . . . . . . 6

2 Economic MPC with extended horizon 9

2.1 Problem setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.2 System description . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.3 EMPC based on an auxiliary controller . . . . . . . . . . . . . 10

2.2 Stability and convergence . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 Practical stability . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.2 Exponential shrinkage . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Asymptotic and transient performance . . . . . . . . . . . . . . . . . 27

2.3.1 Asymptotic performance . . . . . . . . . . . . . . . . . . . . . 28

2.3.2 Transient performance . . . . . . . . . . . . . . . . . . . . . . 30

2.4 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3 Economic MPC with local optimality 45

3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

vi


3.2 Optimal operation of the system . . . . . . . . . . . . . . . . . . . . . 46

3.3 EMPC with the proposed terminal cost . . . . . . . . . . . . . . . . . 51

3.4 Case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4 Economic MPC for scheduled switching operations 60

4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61


4.1.3 A set of auxiliary controllers . . . . . . . . . . . . . . . . . . . 62

4.2 Proposed empc for scheduled switching operations . . . . . . . . . . . 63

4.2.1 Implementation strategy . . . . . . . . . . . . . . . . . . . . . 64

4.2.2 Proposed EMPC design . . . . . . . . . . . . . . . . . . . . . 66

4.2.3 Recursive feasibility . . . . . . . . . . . . . . . . . . . . . . . . 68

4.3 Application to a chemical process . . . . . . . . . . . . . . . . . . . . 70

5 Nonlinear MPC for zone tracking 74

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.2 Problem setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.2.2 System description and control objective . . . . . . . . . . . . 76

5.3 Zone MPC formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.4 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.5 Further discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.5.1 Enlargement of the region of attraction . . . . . . . . . . . . . 83

5.5.2 Handling a secondary economic objective . . . . . . . . . . . . 86

5.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6 Application to an Oilsand Primary Separation Vessel 93

6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

vii


6.1.3 Stability assumption . . . . . . . . . . . . . . . . . . . . . . . 94

6.2 EMPC with terminal cost . . . . . . . . . . . . . . . . . . . . . . . . 95

6.2.1 Properties of the nonlinear controller h(x) . . . . . . . . . . . 95

6.2.2 Proposed EMPC design . . . . . . . . . . . . . . . . . . . . . 97

6.2.3 Stability and performance analysis . . . . . . . . . . . . . . . 99

6.3 Fixed-time implementation . . . . . . . . . . . . . . . . . . . . . . . . 103

6.3.1 Implementation strategy . . . . . . . . . . . . . . . . . . . . . 103

6.3.2 Stability and performance analysis . . . . . . . . . . . . . . . 104

6.4 Application to a primary separation vessel . . . . . . . . . . . . . . . 107

6.4.1 Process description and modeling . . . . . . . . . . . . . . . . 107

6.4.2 EMPC design . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.4.3 Simulation result . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

7 Conclusions and Future Work 120

7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7.2 Future research directions . . . . . . . . . . . . . . . . . . . . . . . . 122

viii

List of Tables

2.1 Shifted asymptotic performance (Jasy−l(xs, us)) and average optimiza-

tion evaluation time (CPU) of the proposed EMPC with N = 1 . . . 40

2.2 Shifted asymptotic performance (Jasy−l(xs, us)) and average optimiza-

tion evaluation time (CPU) of the EMPC without terminal cost. . . . 41

2.3 Transient performance (J30) and optimization evaluation time (CPU)

of the proposed EMPC with terminal cost with N = 1. . . . . . . . . 42

2.4 Transient performance (J30) and optimization evaluation time (CPU)

of the EMPC without terminal cost. . . . . . . . . . . . . . . . . . . 43

3.1 Process parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2 Process parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.1 Model parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.2 Performance under different configurations . . . . . . . . . . . . . . . 73

4.3 Average controller evaluation time . . . . . . . . . . . . . . . . . . . . 73

5.1 The transient performance of the four zone MPC configurations over

30 sampling times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.1 Process variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.2 Model parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.3 Steady-state operating point. . . . . . . . . . . . . . . . . . . . . . . 111

6.4 Transient and steady-state average bitumen recovery rates of the closed-

loop system under the proposed EMPC with terminal cost, the EMPC

in [1], the tracking MPC and the controller h(x) . . . . . . . . . . . 113

ix

6.5 Average bitumen recovery rates of the closed-loop system under (a):

the infinite-time implementation, (b): fixed-time implementation, (c):

alternative fixed-time implementation, and (d): the controller h(x). . 116

x

List of Figures

1.1 A conceptual picture of the MPC scheme . . . . . . . . . . . . . . . . 3

2.1 Closed-loop system state trajectories under the proposed EMPC with

N = 1 and Nh = 1, 5, 10, 15, 20 (solid lines), EMPC without terminal

cost with N = 1, 5, 10, 15, 20 (dashed lines), and the auxiliary

controller h (dash-dotted line). . . . . . . . . . . . . . . . . . . . . . . 39

2.2 Transient performance under the proposed EMPC with N = 1 and

Nh = 1, 5, 10 (solid lines), EMPC without terminal cost with N =

1, 5, 10 (dashed lines), and the auxiliary controller h (dash-dotted line). 40

2.3 State trajectories of the closed-loop system under the proposed EMPC

with N = 1 and Nh = 1, 2, 5 (solid lines), EMPC without terminal

cost with N = 1, 5, 10, 15, 20 (dashed lines), and the auxiliary

controller h (dash-dotted line). . . . . . . . . . . . . . . . . . . . . . . 42

2.4 Stage costs of the closed-loop system under the proposed EMPC with

N = 1 and Nh = 1, 2, 5 (solid lines), EMPC without terminal cost

with N = 1, 5, 10, 15, 20 (dashed lines), and the auxiliary controller

h (dash-dotted line). . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.1 Closed-loop state trajectories under EMPC with the proposed terminal

cost (solid lines), EMPC without terminal cost (dashed lines), and

EMPC with terminal cost for stability [2] (dash-dotted line). . . . . . 57

3.2 Closed-loop input trajectories under EMPC with the proposed terminal

cost (solid lines), EMPC without terminal cost (dashed lines), and

EMPC with terminal cost for stability [2] (dash-dotted line). . . . . . 58

xi

3.3 Closed-loop stage costs under EMPC with the proposed terminal cost

(solid lines), EMPC without terminal cost (dashed lines), and EMPC

with terminal cost for stability [2] (dash-dotted line). . . . . . . . . . 58

4.1 Time flow of Algorithm 1 . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.2 Closed-loop state trajectories of the proposed approach (solid lines),

the approach in [3] (dashed lines) and the auxiliary controllers (dash-

dotted line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.3 Closed-loop input trajectories of the proposed approach (solid line), the

approach in [3] (dashed line) and the auxiliary controllers (dash-dotted

line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.1 Closed-loop state trajectory under controller u = h(x): for any initial

state x ∈ Xh, the trajectory xh(k, x) enters the positive invariant set

XMt,h in Nh steps and is maintained in it thereafter. . . . . . . . . . . . 85

5.2 State and input trajectories of a): a zone MPC that tracks the target

zone Zt (diamonds), b): a zone MPC that tracks the largest control in-

variant set in the target zone ZMt (squares), c): the proposed zone MPC

of Eq. (5.2) (triangles), and d): the proposed zone MPC of Eq. (5.7)

(circles). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.3 State and input trajectories of set-point MPC. . . . . . . . . . . . . . 90

5.4 State and input trajectories of the zone MPC. . . . . . . . . . . . . . 90

5.5 State and input trajectories of set-point MPC. . . . . . . . . . . . . . 90

5.6 State and input trajectories of the zone MPC. . . . . . . . . . . . . . 91

5.7 Bifurcation diagram of system (5.10). The bifurcation diagram shows

the the values of system state x(n) that are visited or approached

asymptotically under a constant u(n). For u ∈ (−0.75,−0.6], the

system has stable steady-state operation. At u = −0.75, steady-state

operation becomes unstable and the diagram bifurcates into two points

which corresponds to a stable periodic operation of period two. As the

value of u decreases, more bifurcation occurs and at some point around

u = −1.4 the system enters a chaotic operating region. The dashed

line corresponds to steady-state operating points . . . . . . . . . . . . 92

xii

6.1 Schematic of the primary separation vessel . . . . . . . . . . . . . . . 107

6.2 Bitumen recovery rate trajectores of the closed-loop system under the

proposed EMPC with terminal cost (dashed line), the EMPC in [1]

(dotted line), the tracking MPC (dash dotted line) and the controller

h(x) (solid line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.3 Froth layer volume trajectores of the closed-loop system under the

proposed EMPC with terminal cost (dashed line), the EMPC in [1]

(dotted line), the tracking MPC (dash dotted line) and the controller

h(x) (solid line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.4 Lyapunov function trajectores of the closed-loop system under the pro-

posed EMPC with terminal cost (dashed line), the EMPC in [1] (dot-

ted line), the tracking MPC (dash dotted line) and the controller h(x)

(solid line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.5 Input trajectories of the closed-loop system under the proposed EMPC

with terminal cost (dashed line), the EMPC in [1] (dotted line), the

tracking MPC (dash dotted line) and the controller h(x) (solid line). . 115

6.6 Bitumen recovery rates of the closed-loop system under the infinite-

time implementation (dashed line), fixed-time implementation(dotted

line), alternative fixed-time implementation (dash dotted line) and the

controller h(x) (solid line). . . . . . . . . . . . . . . . . . . . . . . . . 117

6.7 Froth layer volume trajectories of the closed-loop system under the

infinite-time implementation (dashed line), fixed-time implementation(dotted



6.8 Lyapunov function trajectories of the closed-loop system under the

infinite-time implementation (dashed line), fixed-time implementation(dotted



6.9 Input trajectories of the closed-loop system under the infinite-time

implementation (dashed line), fixed-time implementation(dotted line),

alternative fixed-time implementation (dash dotted line) and the con-

troller h(x) (solid line). . . . . . . . . . . . . . . . . . . . . . . . . . . 118

xiii

Chapter 1

Introduction

1.1 Motivation

Model predictive control (MPC) is a rich fruit in the branch of model control theory

which is rooted in computer science and informatics. Ever since its origin in chemical

plants in the late 1970s, both the industrial applications and theory development of

MPC have been booming. A survey [4] conducted in mid-1999 showed more than

4500 industrial MPC applications prior to the new millennium and indicated a rapid

annual growth of approximately 20%. Presently, not only has MPC become the stan-

dard advanced control technique in chemical engineering, it is also being applied to

various areas in mechanical engineering, electrical engineering, automotive engineer-

ing, aerospace engineering and so on. The landscape of MPC is shaped by three major

aspects: theory, computation and application. To a large extent, early development

of MPC theory was pushed by its successful industrial applications. The progresses in

MPC theory in return stimulated more applications which forms a positive feedback

loop. Since the implementation of MPC requires online or real-time optimization,

all MPC designs and applications have to reckon with the computational burden.

This entails either the use of more powerful optimization algorithms or the design of

computationally efficient MPC.

Traditionally, in the process industry, MPC is designed for tracking set-point or

reducing variations in crucial process variables. The huge success of MPC in indus-

trial applications owes much to its ability to optimally handle process constraints and

interactions. However, in many applications, especially at a higher level of decision-

making, the objectives are often economic-oriented and may be different from the

1

classical control objective of set-point tracking. For example, some typical objec-

tives include the maximization of operation profit, the minimization of energy con-

sumption, or the maximization of a certain product. These economic objectives may

not be seamlessly translated into set-point tracking objectives. It is possible that

non steady-state operation, such as periodic operation, yields superior performance.

These considerations motivate the recent development of a more general form of MPC

called economic MPC or EMPC. EMPC optimizes general cost functions that are di-

rectly linked to the economic metrics (profit, efficiency, sustainability) of the plant.

This promising approach, which has great potential to improve the dynamic perfor-

mance during transient processes, is still a relatively new research area. New theories,

algorithms and tools are being developed for EMPC.

In this thesis, we propose a computationally efficient EMPC design which is based

on a well-known methodology in conventional MPC — separation of the control and

prediction horizon. Our objectives are to systematically analyze the stability and

performance of the general EMPC scheme with extended horizon, and to explore its

extensions and applications to several specific scenarios.

1.2 Research overview

1.2.1 A brief review of MPC

Technically, MPC refers to a control methodology that repeatedly solves online a

finite-horizon open-loop optimal control problem in a receding horizon fashion. Fig-

ure 1.1 shows a conceptual picture of the MPC scheme. At a sampling time k, the

MPC algorithm computes a future input trajectory by optimizing the predicted tra-

jectories over a finite prediction horizon with the initial state x(k). The first element

of the input sequence is then injected to the plant. Upon availability of an updated

state measurement (or estimation) at the subsequent sampling time k + 1, the MPC

algorithm is repeated with the prediction horizon shifted one step further.

MPC differs from the classical optimal control mainly in the way the optimal con-

trol problem is solved. In classical optimal control such as LQR control, an optimal

feedback control law is obtained offline whereas in MPC, an input sequence is calcu-

lated online based on the current system state. The former requires the solution of the

2

FuturePast

Reference trajectory/Set-point

Predicted trajectories

Predicted horizon

… ...

Computed future input trajectory

k

( )x k

1k k N

Figure 1.1: A conceptual picture of the MPC scheme

Hamilton-Jacobi-Bellman equation which can be very difficult or merely impossible

to obtain for generic nonlinear systems. The latter amounts to solving a finite di-

mensional mathematical programming problem in real-time which is becoming more

and more viable thanks to the development of computer hardware/software and the

advances in optimization algorithms.

An essential feature that shapes the landscape of MPC theory and design is that

the open-loop optimal control problem be solvable in a short period of time (as com-

pared to the dynamics of the process). For one thing, online optimization necessitates

the use of a finite horizon which creates a discrepancy between MPC theory and clas-

sical infinite-horizon optimal control. This discrepancy has spawned considerable

research interests in the design and analysis of MPC with nominal stability guaran-

tee. Consensus was reached in the milestone paper [5] in which the use of the value

function as Lyapunov function and several ‘ingredients’ for nominal stability of MPC

are formalized. Since then, more insights into the inherent stability of MPC have

been revealed by leveraging the dynamic programming principle [6], [7], [8], [9], [10].

For another thing, computational efficiency of MPC will remain a constant theme

throughout the development of MPC. This is a twofold mission. On the one hand, de-

velopment of fast optimization algorithms serve as the most straightforward impetus

for the computational efficiency of MPC. The majority of industrial MPC optimiza-

3

tion problems are formulated as quadratic programming (QP) problems which can

be efficiently solved using standard algorithms such as the interior-point methods

[11, 12], the active-set method [13], and fast gradient methods [14]. Recent advances

in nonlinear programming has also made online implementation of MPC to nonlin-

ear, large-scale systems possible, promoting an integrated MPC scheme for plant-wide

real-time optimization [15]. On the other hand, significant research efforts have been

made to reduce the computational load of MPC. These include heated research areas

such as (i) distributed MPC (see [16] and the references therein), where a centralized

MPC is decomposed into several subsystem MPCs which communicate and cooper-

ate; and (ii) explicit MPC (e.g., [17], [18]), where an explicit feedback control law

is obtained offline by solving a parametric programming problem or obtaining its

approximate solution.

1.2.2 Separation of control and prediction horizon

One of the most commonly adopted approach to improve the computational efficiency

of MPC is to separate the control horizon and the prediction horizon. The idea is to

extend the prediction horizon beyond the control horizon by employing an auxiliary

control law which asymptotically stabilizes the desired set-point. In this way, the

optimization horizon of MPC can be increased without increasing the number of

decision variables (i.e. the number of free inputs). As a matter of fact, separation

between the control horizon and the prediction horizon arises along early versions of

MPC and is well embraced in industrial MPC [4]. With the development of MPC

theory, however, separation between the control horizon and the prediction horizon

is gradually being phased out in the context of linear MPC. It was shown in [19] that

the constrained infinite-horizon LQR can be formulated as a finite-dimensional QP.

Specifically, the infinite-horizon tail of the LQR objective function can be lumped

together as a terminal cost function which is the value function of an unconstrained

infinite-horizon LQR. Under this design, if the control horizon is sufficiently large

such that the predicted state at the end of the control horizon can be steered to the

origin by the unconstrained infinite-horizon LQR without violating state and input

constraints, then the MPC is equivalent to the optimal constrained infinite-horizon

LQR controller. The availability of the locally optimal terminal cost function as well

4

as the efficient QP algorithms mentioned above make the separation between control

and prediction horizon less appealing for linear MPC designs.

However, for nonlinear systems, the infinite-horizon cost-to-go is in general not

available because the solution to the corresponding HJB equation may not exist. A

practical remedy is to use a control Lyapunov function (CLF) which provides an

incremental upper bound on the stage cost [20, 21, 7] as the terminal cost function.

NMPC designed this way has guaranteed stability but can be overly conservative,

especially when the terminal cost is much larger than the actual infinite-horizon cost-

to-go. Theoretically, one could use a large control horizon to achieve asymptotic

stability as well as near-optimal performance [6, 8, 22]. But the use of a large control

horizon poses serious challenge to the computational efficiency of MPC. An ideal

trade-off is proposed in [23], where a locally optimal control law is utilized to extend

the prediction horizon of the NMPC design. The NMPC in [23] is capable of achieving

enlarged stability region and locally optimal performance without relying on a large

control horizon. These desired features of NMPC design are only possible via the

separation of control and prediction horizons. It is therefore safe to say that the

methodology to separate control and prediction horizon bears its indispensable merits

in the context of NMPC design.

1.2.3 Economic MPC

With the maturing of MPC theory, recent development of MPC has seen an explo-

ration into economic-oriented model predictive control. This line of research originates

from [24] where unreachable set-points are considered in the conventional tracking

MPC scheme. The authors showed that tracking unreachable set-point results in

improved transient performance as compared to the conventional MPC that tracks

reachable set-point. These results simulated research efforts into a novel form of

MPC called economic MPC (EMPC). In EMPC, the quadratic-type cost functions

used in conventional MPC are replaced with general economic cost functions that

are not necessarily positive-definite with respect to the optimal steady-state opera-

tion. Consequently, standard stability analysis techniques to use the value function

of conventional MPC as a Lyapunov function is no longer viable. In fact, steady-state

operation may not even be the economically optimal operation for EMPC. It has been

5

realized that dissipativity plays an important role in characterizing the optimality of

steady-state operation as well as establishing the stability of EMPC [25, 26, 27].

Different EMPC designs have been proposed which stem from the conventional

NMPC designs. For example, EMPC with point-wise terminal constraint [28], EMPC

with terminal cost [2, 29], EMPC with Lyapunov-based constraint [30, 1]. These

EMPC designs also suffer the problems encountered in the conventional NMPC design

— they could be overly conservative or computationally demanding. In another

line of research [10, 31], EMPC without terminal conditions is studied. This line of

research reveals some intrinsic properties of EMPC. It is shown that under certain

controllability and dissipativity conditions, near-optimal performance can be achieved

if a sufficiently large control horizon is used. However, a large control horizon could

make online implementations of the EMPC design computationally impractical. It is

thus natural to also resort to the separation of control and prediction horizon in the

context of EMPC to improve its computational efficiency.

1.3 Contributions and thesis outline

The rest of the thesis is organized as follows:

In Chapter 2, we propose the basic EMPC formulation with extended prediction

horizon based on an auxiliary controller. The extension of the prediction horizon is

realized by employing a terminal cost which characterizes the economic performance

of the auxiliary controller over a finite prediction horizon. The proposed EMPC

design is easy to construct and computationally efficient. We analyze the stability

and performance of the proposed EMPC design with special attention paid to the

impact of the extended horizon. It is shown that for strictly dissipative systems

satisfying mild assumptions, a finite terminal horizon is sufficient to guarantee the

convergence and performance of the EMPC to be approximately upper-bounded by

that of the auxiliary controller.

In Chapter 3, we design a terminal cost for economic model predictive control

(EMPC) which preserves local optimality. From the results in Chapter 2, the perfor-

mance of EMPC is upper-bounded by the auxiliary controller. A very natural ques-

tion to ask is whether it is possible to find a locally optimal control law and whether

6

the corresponding infinite-horizon cost to go or its approximation can be found. We

first show, based on the strong duality and second order sufficient condition (SOSC)

of the steady-state optimization problem, that the optimal operation of the system

is locally equivalent to an infinite-horizon LQR controller. The proposed terminal

cost is constructed as the value function of the LQR controller plus a linear term

characterized by the Lagrange multiplier associated with the steady state constraint.

EMPC with the proposed terminal cost is stabilizing with an appropriately chosen

control horizon, and preserves the local optimality of the LQR controller. Simulation

results of an isothermal CSTR verify our analysis.

In Chapter 4, we extend the proposed EMPC design to control systems with sched-

uled switching operations. The proposed EMPC scheme takes advantage of a set of

auxiliary controllers that locally stabilizes the optimal steady state of each operating

mode. In the proposed approach, EMPC operations are divided into two phases — an

infinite-time operation phase and a mode transition phase, depending on the current

sampling time and the scheduled mode switching time. Sufficient conditions to en-

sure recursive feasibility of the proposed EMPC design are established. The proposed

EMPC design is computationally efficient and enjoys enlarged feasibility regions than

the auxiliary controllers. Simulation results of a chemical process example demon-

strate the superiority of our design over existing MPC designs for switched scheduling

operations.

In Chapter 5, we propose a general framework for the design and analysis of

nonlinear model predictive control for zone tracking. Zone tracking objective can be

viewed as a special case of economic objective. The proposed zone MPC penalizes

the distance of the predicted state and input trajectories to a desired target zone

which is not necessarily positive invariant. We resort to the invariance principle

and develop invariance-like theorem which is suitable for stability analysis of zone

control. It is proved that under the zone MPC design, the system converges to the

largest control invariant subset of the target zone. Further discussions are made on

enlargement of the region of attraction by employing an auxiliary controller as well

as handling a secondary economic objective via a second-step economic optimization.

Two numerical examples are used to demonstrate the superiority of zone control over

set-point control and the efficacy of the zone MPC design.

7

In Chapter 6, we apply the proposed EMPC algorithm to an oilsand primary sep-

aration vessel (PSV). We show how previously developed EMPC design and analysis

results in the context of discrete-time system can be extended to continuous-time

systems where the issue of sampling needs to be addressed.

Chapter 7 summaries the contributions of this work and discusses future research

directions.

8

Chapter 2

Economic MPC with extendedhorizon

In this chapter, we propose the basic EMPC formulation with extended prediction

horizon based on an auxiliary controller. The system and EMPC formulation are

set up in Section 2.1. Section 2.2 addresses the stability and convergence of the

EMPC design. Practical stability of the proposed EMPC design is established for

strictly dissipative systems satisfying mild assumptions. Under stronger conditions,

the shrinkage of the practical stability region is shown to be exponential with respect

to the increase of the terminal horizon. For a special type of systems which satisfy a

further condition on the storage function (including conventional MPC with quadratic

cost), exponential stability can be achieved. Interestingly, the same result for this type

of systems may not be achieved by an EMPC without terminal condition. Section 2.3

discusses the asymptotic and transient performance of the EMPC design. Results

on the asymptotic performance of the proposed EMPC design for general nonlinear

systems are provided first. Stronger results on the transient performance of the EMPC

design for strictly dissipative systems are derived subsequently, based on different

stability conditions from Section 2.2. Two numerical examples are used to verify our

results in Section 2.4. Finally, we conclude our results in Section 2.5.

9

2.1 Problem setup

2.1.1 Notation

Throughout this work, the operator | · | denotes the Euclidean norm of a scalar or a

vector. The symbol ‘\’ denotes set substraction such that A \ B := x ∈ A, x /∈ B.

The symbol Br(xs) denotes the open ball centered at xs with radius r such that

Br(xs) := x : |x − xs| < r. A continuous function α : [0, a) → [0,∞) is said

to belong to class K if it is strictly increasing and satisfies α(0) = 0. A class K

function α is called a class K∞ function if α is unbounded. A continuous function

σ : [0,∞)→ [0, a) is said to belong to class L if it is strictly decreasing and satisfies

limx→∞

σ(x) = 0. A continuous function β : [0, a) × [0,∞) → [0,∞) is said to belong

to class KL if for each fixed r, β(r, s) belongs to class L, and for each fixed s, β(r, s)

belongs to class K.

2.1.2 System description

We consider a class of nonlinear systems which can be described by the following

discrete state-space model:

x(k + 1) = f(x(k), u(k)) (2.1)

where x ∈ Rnx denotes the state vector and u ∈ Rnu denotes the control input

vector. The system state and input are subject to the constraints x ∈ X and u ∈ U

respectively, where X ⊂ Rnx and U ⊂ Rnu are compact sets. We assume that there

exists an optimal steady state (xs, us) that uniquely solves the following steady-state

optimization problem:(xs, us) = arg min

x,ul(x, u)

s.t. x = f(x, u)x ∈ Xu ∈ U

(2.2)

where l(x, u) : X× U→ R is the economic stage cost function.

2.1.3 EMPC based on an auxiliary controller

It is assumed that there exists an auxiliary explicit controller u = h(x) which renders

xs asymptotically stable with us = h(xs) while satisfying the input constraint for

10

all x ∈ Xf , where Xf ⊆ X is a compact set containing xs in its interior. It is

also assumed that the region Xf is forward invariant under the controller u = h(x).

Namely, f(x, h(x)) ∈ Xf holds for all x ∈ Xf . We use xh(k, x) to denote the closed-

loop state trajectory under the controller h at time instant k with the initial state

xh(0, x) = x. The above assumptions imply that there exists a class KL function βx

such that:|xh(k, x)− xs| ≤ βx(|x− xs|, k)

xh(k, x) ∈ Xf

h(xh(k, x)) ∈ U(2.3)

for all k ≥ 0 and x ∈ Xf .

Our EMPC design takes advantage of the auxiliary controller h(x) to extend the

prediction horizon. Specifically, this is implemented by employing the following ter-

minal cost Vf (x,Nh), which characterizes the economic performance of the controller

h(x) for Nh steps with the initial state x ∈ Xf :

Vf (x,Nh) =

Nh−1∑k=0

l(xh(k, x), h(xh(k, x)))

At a time instant n, our EMPC design is formulated as the following optimization

problem P(n):

minu(0),u(1),...,u(N−1)

N−1∑k=0

l(x(k), u(k)) + Vf (x(N), Nh) (2.4a)

s.t. x(k + 1) = f(x(k), u(k)), k = 0, ..., N − 1 (2.4b)

x(0) = x(n) (2.4c)

x(k) ∈ X, k = 0, ..., N − 1 (2.4d)

u(k) ∈ U, k = 0, ..., N − 1 (2.4e)

x(N) ∈ Xf (2.4f)

where x(k) denotes the predicted state trajectory, x(n) is the state measurement at

time instant n. The optimal solution to the above optimization problem is denoted

as u∗(k|n), k = 0, ..., N − 1. The corresponding optimal state trajectory is x∗(k|n),

k = 0, ..., N . The manipulated input of the closed-loop system under the EMPC at a

time instant n is: u(n) = u∗(0|n). At the next sampling time n+ 1, the optimization

of Eq. (2.4) is re-evaluated. The feasibility region of the optimization problem of

11

Eq. (2.4) is denoted by XN := x(n) : P(n) is feasible. XN is forward invariant

under the EMPC design of Eq. (2.4) due to the forward-invariance of the terminal

region Xf . In other words, the EMPC design is recursively feasible.

It is well understood that conventional MPC with an infinite-horizon terminal

cost (i.e., Vf (x,Nh) with Nh →∞) is stabilizing [21]. Similar result has been estab-

lished in the framework of EMPC for systems satisfying a strong duality condition

[28]. However, the framework of Eq. (2.4) with an infinite terminal horizon is not

implementable unless an analytical form of Vf (x,∞) is available, which is difficult

to construct for generic nonlinear systems, if it exists at all. Thus it is natural to

ask whether the MPC/EMPC of Eq. (2.4) with a finite terminal horizon Nh will

stabilize the system or provide satisfactory performance. The issue has been partly

addressed for conventional MPC in [23]. It is shown in [23] that for any stabilizing

linear controller h(x), there is always a finite prediction horizon such that the MPC

with extended prediction horizon is stabilizing. Most impressively, if h(x) is chosen

as the locally optimal linear quadratic (LQ) controller, then the MPC behaves like

the LQ controller when x(n) is close to xs for sufficiently large Nh, regardless of N .

This locally near optimal behaviour cannot be achieved by other terminal cost designs

where nonlinearity is approximately handled to make the terminal cost compatible

with the stage cost (e.g., [20, 2]). To our knowledge, so far there are no results ad-

dressing the finite terminal horizon in the framework of EMPC. This work fills this

gap. Our analysis is carried out in a general setting where we do not assume con-

tinuous differentiability of the system — a condition under which a linear stabilizing

auxiliary controller and the corresponding forward invariant set can be readily con-

structed (see e.g., [32] (pp.136-137) ). This allows our analysis to be applicable to a

broader class of nonlinear systems. Interested readers may refer to [33, 34, 35] and

references therein for some existing nonlinear controller design techniques. In the

following we will proceed with our discussions assuming that the auxiliary controller

h(x) is known while we focus on the impact of a finite terminal horizon Nh.

12

2.2 Stability and convergence

We restrict our attentions to systems that are strictly dissipative with respect to

the economic cost functions. Systems of this type are optimally operated at steady

state [25, 26, 27]. Since the optimal steady state is not necessarily a minimizer of

the economic stage cost, EMPC with a finite horizon in general cannot stabilize the

optimal steady state. We will first establish practical stability of the EMPC design

with respect to the terminal horizon Nh in a general setting. Then under a set of

stronger conditions, we show that the shrinkage of the practical stability region can

be exponential. Finally, we show that for a special type of systems satisfying a further

condition on the storage function (including conventional MPC with quadratic cost),

exponential stability can be achieved.

2.2.1 Practical stability

Definition 1 (Strictly dissipative systems) The system of Eq. (2.1) is strictly dissi-

pative with respect to the supply rate s : X×U→ R if there exists a storage function

λ : X→ R and a class K function αl such that the following holds for all x ∈ X and

u ∈ U:

λ(f(x, u))− λ(x) ≤ s(x, u)− αl(|x− xs|)

Assumption 1 (Strict dissipativity) The system of Eq. (2.1) is strictly dissipative

with respect to the supply rate s(x, u) = l(x, u)− l(xs, us)

Assumption 2 (Continuity) The functions f and l are continuous on X × U, h is

continuous on Xf , λ is continuous on X.

Assumption 3 (Bounded supply under h(x)) There exists a class K∞ function αh

such that the accumulated supply rate s(x, u) = l(x, u)− l(xs, us) under the auxiliary

controller h(x) is bounded such that

Nh−1∑k=0

l(xh(k, x), h(xh(k, x))−Nhl(xs, us) ≤ αh(|x− xs|)

for all x ∈ Xf and Nh ≥ 1.

13

Lemma 1 (c.f [2, 36]) Let V (x) be a bounded function defined on a closed set X ⊂

Rnx containing xs. If V (x) is continuous at xs with V (xs) = 0, then there exists a

class K∞ function α such that:

V (x) ≤ α(|x− xs|), ∀x ∈ X (2.5)

Remark 1 In the light of Lemma 1 and based on Assumption 2, the definition of

strictly dissipative systems in Definition 1 which employs a class K function is equiv-

alent to the definition made in [25] where a positive-definite function is employed.

Similarly, if the continuity Assumption 2 holds, then Assumption 3 is equivalent to as-

suming that the accumulated supply rate under the auxiliary controller h(x) is bounded

from above.

To proceed, let us define the rotated cost function l(x, u) as follows:

l(x, u) = l(x, u)− l(xs, us) + λ(x)− λ(f(x, u)) (2.6)

Based on Assumption 1, the rotated economic cost function l(x, u) is bounded from

below by:

l(x, u) ≥ αl(|x− xs|) (2.7)

for all x ∈ X, u ∈ U. The rotated terminal cost is defined accordingly as:

Vf (x,Nh) =

Nh−1∑k=0

l(xh(k, x), h(xh(k, x)))

And we define the rotated optimization problem P(n) as follows:

minu(0),u(1),...,u(N−1)

N−1∑k=0

l(x(k), u(k)) + Vf (x(N), Nh) + λ(xh(Nh, x(N))− λ(xs)

s.t. (2.4b)-(2.4f)

(2.8)

Lemma 2 The optimal solutions to the optimization problems P(n) of Eq. (2.4) and

P(n) of Eq. (2.8) are identical.

Proof. Let us use VN,Nh(x(n), u) to denote the objective function of the rotated

optimization problem P(n). It can be shown that:

VN,Nh(x(n), u) = VN,Nh(x(n), u) + (N +Nh)l(xs, us)− λ(x(n)) + λ(xs) (2.9)

14

Since P(n) and P(n) have the same constraint set and their objective functions only

differ by constant terms (N+Nh)l(xs, us), λ(x(n)) and λ(xs) which are all independent

on the decision variables u(0), ..., u(N − 1), they have the same optimal solution.

The equivalence of solutions between the original problem P(n) and the rotated

problem P(n) allows us to carry out stability analysis of the closed-loop system based

on P(n). In the following, we will first define a so-called relaxed practical Lyapunov

function as an analysis tool for practical stability and then show that the optimal

objective function value of P(n) is a relaxed practical Lyapunov function.

Definition 2 (Relaxed practical Lyapunov function) A function V (x) : S → R de-

fined on a forward-invariant set S is called a relaxed practical Lyapunov function with

respect to positive scalars δ1, δ2, δ3, if there exist class K∞ functions α1, α2, α3 such

that the closed-loop state trajectory x(k), k ≥ 0 satisfies:

α1(|x(k)− xs|)− δ1 ≤ V (x(k)) ≤ α2(|x(k)− xs|) + δ2 (2.10a)

V (x(k + 1)) ≤ V (x(k))− α3(|x(k)− xs|) + δ3 (2.10b)

Theorem 1 If there exists a relaxed practical Lyapunov function V (x) on a forward-

invariant set S with positive scalars δ1, δ2, δ3 and K∞ functions α1, α2, α3 as defined

in Eq. (2.10) for the closed-loop system of Eq. (2.1), and if Br(xs) ⊂ S where r =

α−11 (α2(α−1

3 (δ3)) + δ1 + δ2 + δ3), then there exists a KL function β such that the

following holds for all x(0) ∈ S, k ≥ 0:

|x(k)− xs| ≤ maxβ(|x(0)− xs|, k), r

Proof. In this proof, we first show that if V (x) ≥ α2(α−13 (δ3)) + δ2 + δ3, it keeps

decreasing until it reaches V (x) < α2(α−13 (δ3)) + δ2 + δ3. Then we show that the

α2(α−13 (δ3)) + δ2 + δ3 level set of V (x) is forward invariant. Finally, we construct β

and r based on these results.

First, if V (x(k)) ≥ α2(α−13 (δ3)) + δ2 + δ3, from Eq. (2.10a) we have: α2(α−1

3 (δ3)) +

δ3 ≤ α2(|x(k) − xs|), substituting the above into Eq. (2.10b), the following can be

obtained:

V (x(k + 1)) ≤ V (x(k))− α3(α−12 (α2(α−1

3 (δ3))) + δ3) + δ3

15

Since δ3 > 0, there exists a positive scalar ε > 0 such that

α3(α−12 (α2(α−1

3 (δ3))) + δ3) = α3(α−12 (α2(α−1

3 (δ3)))) + ε = δ3 + ε

which gives: V (x(k + 1))− V (x(k)) ≤ −ε. Thus, the following holds for all V (x(0))

and V (x(k)) ≥ α2(α−13 (δ3)) + δ2 + δ3:

V (x(k)) ≤ V (x(0))− kε

Taking into account that V (x) is bounded on S, the above implies that V (x) decreases

to α2(α−13 (δ3)) + δ2 + δ3 in finite time.

Second, we show that the α2(α−13 (δ3))+δ2+δ3 level set of V (x) is forward invariant.

That is, V (x(k + 1)) < α2(α−13 (δ3)) + δ2 + δ3 if V (x(k)) < α2(α−1

3 (δ3)) + δ2 + δ3. We

consider two cases:

(1) |x(k)− xs| ≥ α−13 (δ3). In this case, from Eq. (2.10b):

V (x(k + 1)) ≤ V (x(k))− α3(α−13 (δ3)) + δ3 = V (x(k)) < α2(α−1

3 (δ3)) + δ2 + δ3

(2) |x(k)− xs| < α−13 (δ3). In this case, from Eq. (2.10a):

V (x(k)) ≤ α2(|x(k)− xs|) + δ2 < α2(α−13 (δ3)) + δ2

Substituting the above into Eq. (2.10b):

V (x(k + 1)) < α2(α−13 (δ3)) + δ2 − α3(|x(k)− xs|) + δ3 < α2(α−1

3 (δ3)) + δ2 + δ3

From the above results, the following holds for all x(0) ∈ S:

V (x(k)) ≤ maxV (x(0))− kε, α2(α−13 (δ3)) + δ2 + δ3

Using Eq. (2.10a) and the above, the following can be obtained:

α1(|x(k)−xs|) ≤ maxα2(|x(0)−xs|)+δ1 +δ2−kε, α2(α−13 (δ3))+δ1 +δ2 +δ3 (2.11)

Let us define β:

β(|x(0)− xs|, k) =|x(0)− xs|α−1

3 (δ3)maxα2(|x(0)− xs|) + δ1 + δ2 − kε, 0

16

It can be checked that β belongs to class KL by definition, and that Eq. (2.11) still

holds with the replacement:

α1(|x(k)− xs|) ≤ maxβ(|x(0)− xs|, k), α2(α−13 (δ3)) + δ1 + δ2 + δ3

To verify the above, note that when |x(0) − xs| < α−13 (δ3), the first term on the

right-hand-side of Eq. (2.11) is smaller than the second term; And when |x(0)−xs| ≥

α−13 (δ3), β(|x(0)− xs|, k) ≥ α2(|x(0)− xs|) + δ1 + δ2 − kε.

Theorem 1 is thus proved with β(|x(0) − xs|, k) = α−11 (β(|x(0) − xs|, k)) and

r = α−11 α2(α−1

3 (δ3)) + δ1 + δ2 + δ3).

Theorem 1 characterizes the practical stability of the system on a forward-invariant

set S. Specifically, the system state is driven into an open ball Br(xs) in finite time

and maintained in it thereafter. Let us use V ∗N,Nh(x(n)) to denote the optimal objec-

tive function value of P(n). In the following, we show that under the EMPC design,

V ∗N,Nh(x(n)) is a relaxed practical Lyapunov function on XN , with the corresponding

scalars δ1, δ2, δ3 all being class L functions of Nh.

Lemma 3 If Assumption 2 holds, then there exists a class KL function βl such that:

|l(xh(k, x), h(xh(k, x))− l(xs, us)| ≤ βl(|x− xs|, k) (2.12)

for all x ∈ Xf .

Proof. Based on the continuity of l(·) and h(·) and the fact that us = h(xs),

|l(x, h(x)) − l(xs, us)| is bounded and continuous at 0. Applying Lemma 1, there

exists a class K∞ function α such that

|l(x, h(x))− l(xs, us)| ≤ α(|x− xs|)

for all x ∈ Xf . Using xh(k, x) to replace x and from Eq. (2.3), the following holds for

all x ∈ Xf :

|l(xh(k, x), h(xh(k, x)))− l(xs, us)| ≤ α(|xh(k, x)− xs|) ≤ α(βx(|x− xs|, k))

By definition, α(βx(|x − xs|, k)) belongs to class KL with respect to |x − xs| and k.

Let βl(|x− xs|, k) = α(βx(|x− xs|, k)), Eq. (2.12) is obtained.

17

Lemma 4 If Assumption 2 holds, then there exists a class KL function βλ such that:

|λ(xh(k, x))− λ(xs)| ≤ βλ(|x− xs|, k) (2.13)

for all x ∈ Xf .

Proof. The proof is similar to the proof of Lemma 3 and is omitted here.

Before presenting the final result in Theorem 2, we still need to establish an upper

bound for the auxiliary optimization problem in Lemma 5 below, which will then be

used to bound V ∗N,Nh(x(n)).

Lemma 5 If Assumptions 1-3 hold, then there exists a class K∞ function αv such

that:

V rN,Nh

(x(n)) ≤ αv(|x(n)− xs|) (2.14)

for all x(n) ∈ XN , where the function V rN,Nh

(x(n)) is the optimal objective function

value of the following optimization problem R(n):

V rN,Nh

(x(n)) = minu(0),u(1),...,u(N−1)

N−1∑k=0

l(x(k), u(k)) + c(x(N), Nh)

s.t. (2.4b)− (2.4f)

(2.15)

Proof. In this proof, we show that Lemma 1 can be applied to V rN,Nh

(x(n)). It can

be verified that V rN,Nh

(xs) = 0. Also, V rN,Nh

(x(n)) is continuous at x(n) = xs because

of Assumption 2.

We show next that V rN,Nh

(x(n)) is bounded on XN ⊆ X. Since l(x, u) is bounded

on X×U, the first term in the objective function of Eq. (2.15) is bounded for a given

control horizon N . Based on the definition of l, the rotated terminal cost c(x(N), Nh)

can be equivalently written as:

c(x(N), Nh) =

Nh−1∑k=0

l(xh(k, x(N)), h(xh(k, x(N))) + λ(x(N))− λ(xh(Nh, x(N))

Based on Assumption 3,Nh−1∑k=0

l(xh(k, x(N)), h(xh(k, x(N))) is bounded by αh(x(N)),

the terms λ(x(N)) and λ(xh(Nh, x(N)) are all bounded on Xf . The rotated terminal

cost is thus bounded on Xf , which further implies that V rN,Nh

(x(n)) is bounded on

XN . Applying Lemma 1, there exists a class K∞ function αv such that Eq. (2.14)

holds.

18

Let us define the size of the terminal set:

dmax := max|x− xs| : x ∈ Xf (2.16)

The practical stability of the EMPC design is summarized in the following theorem.

Theorem 2 Consider the system of Eq. (2.1) in closed-loop under the EMPC of

Eq. (2.4), if Assumptions 1-3 hold, then there exist class KL functions βn and βr

such that the following holds for all x(0) ∈ XN and n ≥ 0:

|x(n)− xs| ≤ maxβn(|x(0)− xs|, n), βr(dmax, Nh)

Proof. We will first establish V ∗N,Nh(x(n)) as a relaxed practical Lyapunov function

on XN such that:

αl(|x(n)− xs|)− βλ(dmax, Nh) ≤ V ∗N,Nh(x(n)) ≤ αv(|x(n)− xs|) + βλ(dmax, Nh)

(2.17a)

V ∗N,Nh(x(n+ 1)) ≤ V ∗N,Nh(x(n))− αl(|x(n)− xs|) + βl(dmax, Nh)

(2.17b)

Once the above is established, applying Theorem 1, there exist βn and r such that

|x(n)− xs| ≤ maxβn(|x(0)− xs|, n), r

where r = α−1l

(αV (α−1

l (βl(dmax, Nh))) + 2βλ(dmax, Nh) + βl(dmax, Nh))

is the class

KL function βr(dmax, Nh).

Left half of Eq. (2.17a). From the definition, V ∗N,Nh(x(n)) can be written in

terms of the corresponding optimal state and input trajectories x∗(k|n) and u∗(k|n)

as follows:

V ∗N,Nh(x(n)) =N−1∑k=0

l(x∗(k|n), u∗(k|n)) + c(x∗(N |n), Nh) + λ(xh(Nh, x∗(N |n)))− λ(xs)

(2.18)

Based on Eq. (2.7), we have:

N−1∑k=0

l(x∗(k|n), u∗(k|n)) + c(x∗(N |n), Nh) ≥ l(x(n), u∗(0|n)) ≥ αl(|x(n)− xs|) (2.19)

Using Lemma 4, and taking into account the definition of dmax, the following holds:

|λ(xh(Nh, x∗(N |n)))− λ(xs)| ≤ βλ(|x∗(N |n)− xs|, Nh) ≤ βλ(dmax, Nh) (2.20)

19

Combining Eqs. (2.18), (2.19) and (2.20), the left half of Eq. (2.17a) is obtained.

Right half of Eq. (2.17a). We denote the the optimal solution to the opti-

mization problem R(n) of Eq. (2.15) as u∗r(k|n) and the corresponding optimal state

trajectory as x∗r(k|n), k = 0, ..., N−1. Since P(n) and R(n) have the same constraint

set, u∗r(k|n) is also feasible for P(n). The following inequality holds:

V ∗N,Nh(x(n)) ≤ VN,Nh(x(n), u∗r) = V rN,Nh

(x(n)) + λ(xh(Nh, x∗r(N |n)))− λ(xs) (2.21)

Using Lemma 5 and following Eq. (2.20), the right half of Eq. (2.17a) is obtained.

Eq. (2.17b). Since the input sequence:

u(n+ 1) = [u∗(1|n), ..., u∗(N − 1|n), h(x∗(N |n))]

provides a feasible solution for P(n+ 1), the following holds:

V ∗N,Nh(x(n+ 1)) ≤ VN,Nh(x(n+ 1), u(n+ 1))

= V ∗N,Nh(x(n))− l(x(n), u(n)) + λ(xh(Nh + 1, x∗(N |n)))

−λ(xh(Nh, x∗(N |n))) + l(xh(Nh, x

∗(N |n)), h(xh(Nh, x∗(N |n)))

= V ∗N,Nh(x(n))− l(x(n), u(n))− l(xs, us)+l(xh(Nh, x

∗(N |n)), h(xh(Nh, x∗(N |n)))

(2.22)

Using Lemma 3, the following holds:

l(xh(Nh, x∗(N |n)), h(xh(Nh, x

∗(N |n))))− l(xs, us)≤ βl(|x∗(N |n)− xs|, Nh) ≤ βl(dmax, Nh)

(2.23)

Substituting Eqs. (2.23) and (2.7) into Eq. (2.22), Eq. (2.17b) is obtained.

From the above analysis, V ∗N,Nh(x(n)) is a relaxed practical Lyapunov function of

the closed-loop system on XN . Applying Theorem 1, Theorem 2 is proved.

Theorem 2 shows that the system state will be driven into an open ball Br(xs)

in finite time. The radius of the ball is a class KL function r = βr(dmax, Nh), which

implies that for a sufficiently large but finite terminal horizon Nh, the system state

will converge into a small region containing the optimal steady state. In addition, if

the terminal region is small, then the need to use a large terminal horizon Nh to ensure

convergence of the system state is mitigated. In the extreme case where Xf shrinks to

a point xs so that dmax = 0, Theorem 2 reduces to |x(n)− xs| ≤ βn(|x(0)− xs|, n).

This is consistent with the asymptotic stability of EMPC with a point-wise terminal

constraint [28].

20

2.2.2 Exponential shrinkage

In Theorem 2, the decreasing rate of βr(dmax, Nh) with respect to Nh is not char-

acterized. In other words, under Assumptions 1-3, the magnitude of Nh to ensure

a reasonably small size of the practical stability region is “uncontrolled”. In this

subsection, we show that under a set of stronger conditions, the size of the ball

r = βr(dmax, Nh) shrinks exponentially as Nh increases. Moreover, we show that for

a special case which satisfies a further condition on the storage function, exponential

stability can be achieved under the EMPC design with sufficiently large Nh.

Assumption 4 (Exponential stability of h) There exist positive constants a ≥ 1 and

0 < s < 1 such that the following holds for all k ≥ 0 and x ∈ Xf :

|xh(k, x)− xs| ≤ ask|x− xs|

Assumption 5 (Polynomial bounds) There exist positive constants c1, c2, and p such

that the following holds for all x ∈ XN and u ∈ U:

c1|x− xs|p ≤ l(x, u) ≤ c2(|x− xs|p + |u− us|p)

Assumption 6 (Lipschitz continuity) The functions f and l are Lipschitz continuous

on X× U, h is Lipschitz continuous on Xf , λ is Lipschitz continuous on X.

Lemma 6 If Assumptions 1, 4, 5, 6 hold, then there exists a positive constant af

such that the following holds for all x ∈ Xf and Nh ≥ 1:

c(x,Nh) ≤ af |x− xs|p

Proof. Let Lh denotes the Lipschitz constant of h on Xf . From Assumptions 5 and

6, the following holds for all x ∈ Xf :

l(x, h(x)) ≤ c2(1 + Lhp)|x− xs|p

Thus,

c(x,Nh) =

Nh−1∑k=0

l(xh(k, x), h(xh(k, x))) ≤Nh−1∑k=0

c2(1 + Lhp)|xh(k, x)− xs|p

21

Using Assumption 4:

c(x,Nh) ≤Nh−1∑k=0

c2(1 + Lhp)apspk|x− xs|p

≤∞∑k=0

c2(1 + Lhp)apspk|x− xs|p

= c2(1 + Lhp)ap

1

1− sp|x− xs|p

Lemma 6 is thus established with af = c2(1 + Lhp)ap

1

1− sp.

Lemma 7 If Assumptions 1, 4, 5, 6 hold, then there exists a positive constant av

such that the optimal objective function of the auxiliary optimization problem R(n)

is bounded by

V rN,Nh

(x(n)) ≤ av|x(n)− xs|p

for all x(n) ∈ XN and Nh ≥ 1.

Proof. If x(n) ∈ Xf , the input sequence generated by the auxiliary controller h(x):

Uh = [h(xh(0, x(n))), h(xh(1, x(n)))..., h(xh(N − 1, x(n)))] (2.24)

provides a feasible solution to the optimization problem R(n), with the corresponding

objective function c(x,N + Nh). From Lemma 6, V rN,Nh

(x(n)) ≤ c(x,N + Nh) ≤

af |x(n)− xs|p.

Next, we extend the polynomial upper-bound of V rN,Nh

(x(n)) on Xf to XN . We

first note that V rN,Nh

(x(n)) is bounded on XN because all the terms in the objec-

tive function of R(n) are bounded. Specifically, c(x(N), Nh) is bounded because

of Lemma 6;N−1∑k=0

l(x(k), u(k)) is bounded for a given finite prediction horizon N

due to Assumption 5. Let us define the maximum possible value of V rN,Nh

(x(n)) as

V rmax =: maxV r

N,Nh(x(n)) : x(n) ∈ XN. Let us further define dmin := inf|x − xs| :

x ∈ XN \ Xf. Since the terminal region Xf is a compact set containing xs in its

interior, dmin > 0. From the definition of V rmax and dmin, the following holds for all

x(n) ∈ XN \ Xf :

V rN,Nh

(x(n)) ≤ V rmax

dminp |x(n)− xs|p

Let

av = maxaf ,V r

max

dminp (2.25)

22

Proof. Following the similar arguments as in the proof of Theorem 2, it can be shown

that V ∗N,Nh(x(n)) is a relaxed practical Lyapunov function on XN satisfying:

c1|x(n)− xs|p − aλdmaxsNh ≤ V ∗N,Nh(x(n)) ≤ av|x(n)− xs|p + aλdmaxs

Nh

V ∗N,Nh(x(n+ 1)) ≤ V ∗N,Nh(x(n))− c1|x(n)− xs|p + aldmaxsNh

(2.26)

Specifically, the above can be established following the similar arguments as used in

Theorem 2 with αl(|x− xs|) replaced by c1|x− xs|p due to Assumption 5, βλ(dmax, k)

replaced by aλdmaxsk due to Lemma 9, αv|x(n)−xs| replaced by av|x(n)−xs|p because

of Lemma 7, and βl(dmax, k) replaced by aldmaxsk because of Lemma 8. The details

are omitted here for brevity. Applying Theorem 1, Eq. (2.26) implies the existence

of βn and βer with

βer(dmax, Nh) =(dmax

c1

[avc1

al + 2aλ + al]) 1p(s

1p

)Nhwhich decreases to 0 exponentially fast as Nh increases.

Note that in Theorem 2 or Theorem 3, the exponential decreasing of βn(|x(0) −

xs|, n) with respect to n is trivial. This is because for any given Nh, there is always a

finite time instant n∗ such that βn(|x(0)−xs|, n) ≤ βer(dmax, Nh) for all n ≥ n∗, which

means that the effect of the term βn(|x(0)−xs|, n) disappears in finite time. Therefore,

we can always consider the decreasing rate of βn(|x(0)− xs|, n) with respect to n as

exponential. In the following, we show that for a special case where the following

condition on the storage function is satisfied, exponential stability instead of practical

stability can be achieved. That is, βn(|x(0)− xs|, n) decreases to 0 exponentially fast

as n increases and βer(dmax, Nh) ≡ 0.

Assumption 7 (Polynomial bound on λ) There exists a positive constant cλ, such

that the following holds

|λ(x)− λ(xs)| ≤ cλ|x− xs|p

for all x ∈ Xf , where p is defined in Assumption 5.


Eq. (2.4) with x(0) ∈ XN . if Assumptions 1, 4, 5, 6, 7 hold, then there exists a finite

N∗h such that for all Nh ≥ N∗h , xs is exponentially stable.

24

Proof. We prove that for some sufficiently large Nh , V ∗N,Nh(x(n)) is a Lyapunov

function satisfying

c1|x(n)− xs|p ≤ V ∗N,Nh(x(n)) ≤ cv|x(n)− xs|p (2.27a)

V ∗N,Nh(x(n+ 1))− V ∗N,Nh(x(n)) ≤ −c3|x(n)− xs|p (2.27b)

where c1 is defined in Assumption 5, cv and c3 are positive constants.

Left half of Eq. (2.27a) . We show that the left half of Eq. (2.27a) holds for

all Nh ≥ N1 where

N1 = minNh ∈ Z+ : Nh ≥1

p ln sln(

c1

cλap) (2.28)

where Z+ denotes the set of positive integers. Based on Assumption 4 and Assump-

tion 7,

λ(xh(Nh, x∗(N |n)))−λ(xs) ≥ −cλ|xh(Nh, x

∗(N |n))−xs|p ≥ −cλapsNhp|x∗(N |n)−xs|p

Taking into Assumption 5, it can be verified that the following holds for all Nh ≥ N1:

l(x∗(N |n), h(x∗(N |n))) + λ(xh(Nh, x∗(N |n)))− λ(xs)

≥ (c1 − cλapsNhp)|x∗(N |n)− xs|p ≥ 0(2.29)

Substituting the above into Eq. (2.18) and using Assumption 5, the following holds

for all x(n) ∈ XN and Nh ≥ N1:

V ∗N,Nh(x(n)) ≥ l(x∗(0|n), u∗(0|n)) ≥ c1|x(n)− xs|p

Right half of Eq. (2.27a) The proof is similar to that of Lemma 7. We first

show that there exists a positive constant cf such that V ∗N,Nh(x(n)) ≤ cf |x(n)− xs|p

for all x(n) ∈ Xf . Since the input sequence Uh of Eq. (2.24) provides a feasible

solution for P(n), using Lemma 6 and Assumptions 4 and 7, the following holds for

all x(n) ∈ Xf :

V ∗N,Nh(x(n)) ≤ VN,Nh(x(n), Uh)

= c(x(n), N +Nh) + λ(xh(x(n), N +Nh))− λ(xs)

≤ af |x− xs|p + cλaps(N+Nh)p|x(n)− xs|p

≤ af |x− xs|p + cλap|x(n)− xs|p

25

For the last term on the right-hand-side of Eq. (2.30), from Assumption 5:

−l(x(n), u(n)) ≤ −c1|x(n)− xs|p (2.35)

Substituting Eqs. (2.33), (2.34) and (2.35) into Eq. (2.30):

V ∗N,Nh(x(n+ 1))− V ∗N,Nh(x(n))

≤(cλ(1 + sp)apsNhp

cVc1

+ c2(1 + Lhp)apsNhp

cVc1

− c1

)|x(n)− xs|p

(2.36)

Let c3 = −cλ(1 + sp)apsNhpcVc1

− c2(1 + Lhp)apsNhp

cVc1

+ c1, it can be verified that

c3 > 0 for all Nh > N3 (not necessarily positive) where

N3 = − 1

p ln sln((cλ(1 + sp) + c2(1 + Lph))

cVc2

1

)From the above analysis, let

N∗h = minNh ∈ Z+ : Nh ≥ N1, Nh > N3 (2.37)

Then V ∗N,Nh(x(n)) satisfies Eq. (2.27) for all Nh ≥ N∗h and x(n) ∈ XN , which implies

that xs is exponentially stable on XN .

Note that Assumption 7 does not necessarily hold if the conditions for Theorem 3

are satisfied. If 0 < p ≤ 1 in Assumption 5, then Assumption 6 (Lipschitz continuity

of λ) is a sufficient condition for Assumption 7. If p > 1, then Assumption 7 is

stronger than Assumption 6. We will show in the numerical examples in Section 5 that

Assumption 7 is satisfied for only one of the examples. Note also that Assumption 7

is satisfied automatically in conventional MPC where the stage cost l(x, u) is positive-

definite with respect to (xs, us). In this case, we can always define the rotated stage

cost as the original stage cost l(x, u) = l(x, u), with the storage function λ(x) =

0. Thus Theorem 4 applies to conventional MPC with quadratic stage cost. It

is consistent with the existing results for conventional MPC that a finite terminal

horizon is sufficient to guarantee stability [37, 23].

2.3 Asymptotic and transient performance

In this section, we characterize upper bounds on the performance of the EMPC design.

First, we consider the asymptotic performance of the EMPC for systems that are not

27

necessarily dissipative. Then we show that for strictly dissipative systems satisfying

different stability conditions as discussed in Section 3, stronger results on the transient

performance can be obtained. In our analysis, the transient performance of the EMPC

is compared against a benchmark system h∗ which is an augmented version of the

auxiliary controller h(x) with the same feasibility region as EMPC. It is shown that if

the terminal horizon Nh is sufficiently large, then transient performance of the EMPC

is approximately upper-bounded by that of the benchmark system h∗. This implies

that the EMPC either extends the feasibility region of the auxiliary controller or

improves its performance. Moreover, if h(x) is locally optimal on the terminal region

Xf or around the optimal steady state xs, then the local optimality is approximately

preserved by the EMPC if a sufficiently large terminal horizon Nh is used.

2.3.1 Asymptotic performance

A fundamental consideration when it comes to the design of EMPC is whether it

achieves satisfactory asymptotic performance which is defined as follows:

Jasy := limK→∞

sup1

K

K−1∑k=0

l(x(k), u(k))

Ideally, the asymptotic performance of the EMPC design should be no worse than the

optimal steady state operation, i.e., JEMPCasy ≤ l(xs, us). In general, this goal may not

be achieved by the proposed EMPC design with a finite Nh. The following Theorem

shows that near-optimal asymptotic performance can be achieved if Nh is sufficiently

large.


Eq. (2.4) with x(0) ∈ XN . If f , l, h are continuous, then the asymptotic performance

of the system is bounded by:

JEMPCasy ≤ l(xs, us) + βl(dmax, Nh) (2.38)

where βl is defined in Lemma 3 and dmax in Eq. (2.16). Moreover, if Assumptions 4, 6

hold, then there exists a class KL function βel (dmax, Nh) which decreases exponentially

to 0 with respect to Nh such that:

JEMPCasy ≤ l(xs, us) + βel (dmax, Nh) (2.39)

28

Proof. First we prove Eq. (2.38) under the continuity assumptions of the system.

The proof follows the standard approach to construct a feasible solution for P(n+ 1)

by discarding the first entry of the solution of P(n) and implementing the auxiliary

controller h(x) one more step at the end. Specifically, if P(n) is feasible, the input

sequence U(n+1) = [u∗(1|n), ..., u∗(N−1|n), h(x∗(N |n))] provides a feasible solution

for P(n + 1). Based on the principal of optimality V ∗N,Nh(x(n + 1)) ≤ VN,Nh(x(n +

1), U(n+ 1)) and the construction of U(n+ 1), the following can be obtained:

l(x(n), u(n)) ≤ V ∗N,Nh(x(n))−V ∗N,Nh(x(n+1))+l(xh(Nh, x∗(N |n)), h(xh(Nh, x

∗(N |n)))

Using Lemma 3 and taking into account the definition of dmax, the following can be

obtained:

l(x(n), u(n)) ≤ V ∗N,Nh(x(n))− V ∗N,Nh(x(n+ 1)) + l(xs, us) + βl(dmax, Nh) (2.40)

Summing up both sides of the above inequality from n = 0 to K − 1:

K−1∑n=0

l(x(n), u(n)) ≤ V ∗N,Nh(x(0))− V ∗N,Nh(x(K)) +K(l(xs, us) + βl(dmax, Nh)) (2.41)

Dividing both sides by K with K approaching infinity:

limK→∞

sup1

K

K−1∑n=0

l(x(n), u(n))

≤ limK→∞

sup1

K

(V ∗N,Nh(x(0))− V ∗N,Nh(x(K))

)+ l(xs, us) + βl(dmax, Nh)

The left-hand-side of the above is JEMPCasy .Since V ∗N,Nh(x(n)) is bounded on XN for any

finite N and Nh, we have limK→∞

sup 1K

(V ∗N,Nh(x(0)) − V ∗N,Nh(x(K))

)= 0. This proves

Eq. (2.38).

Eq. (2.39) under Assumptions 4 and 6 can be proved following the similar argu-

ments with βl(dmax, Nh) in Eq. (2.40) replaced with alskdmax (because of Lemma 8)

which is a class KL function βel (dmax, Nh) which decreases exponentially to 0 with

respect to Nh.

Note that the result of Theorem 5 is established for general systems that are

not necessarily dissipative. In other words, the closed-loop system state does not

necessarily converge to the optimal steady state. For strictly dissipative systems, the

near optimal asymptotic performance of Theorem 5 will be a direct result of practical

stability. We show in the following that stronger results on the transient performance

can be obtained for strictly dissipative systems.

29

2.3.2 Transient performance

We employ the solution to P(0) with the objective function V ∗N,∞(x(0)) as the bench-

mark system which is denoted as h∗. Speficically, V ∗N,∞(x(0)) =∞∑k=0

l(xh∗(k), uh∗(k))

with xh∗(0) = x(0). The benchmark system h∗ can be viewed as an improved version

of the auxiliary controller h with enlarged feasiblity region or improved performance.

The transient performance of the system over K time steps is denoted as

JK :=K−1∑k=0

(l(x(k), u(k))− l(xs, us)

)(2.42)

In the following, we compare JEMPCK with Jh

∗K given the same initial state x(0).

Remark 2 Note that V ∗N,Nh(·) is bounded and well-defined for Nh →∞ if the condi-

tions in Section 3 hold. The boundedness of VN,Nh(·) can be seen from Eq. (2.17a) or

Eq. (2.26). Essentially, VN,Nh(·) is bounded from below because of the strict dissipa-

tivity Assumption 1, and is bounded from above because of Assumption 3. In the case

of exponential shrinkage, Assumptions 4 and 5 are sufficient conditions for Assump-

tion 3. Our analysis based on the rotated problem P(n) remains valid even though the

original objective function VN,Nh(·) may grow unbounded as Nh → ∞. In this case,

the result of Lemma 2 can be recovered by subtracting l(xs, us) from the original stage

cost l(x, u).

Lemma 10 If the conditions of Theorem 2 are satisfied, then there exists a class KL

function βx∗ such that:

|xh∗(k)− xs| ≤ βx∗(|x(0)− xs|, k) (2.43)

Moreover, if the conditions of Theorem 3 are satisfied, then the following holds:

|xh∗(k)− xs| ≤ ask−N(avc1

)1p |x(0)− xs| (2.44)

where a and s are defined in Assumption 4, c1 and p are defined in Assumption 5, av

is defined in Lemma 7.

Proof. First, we construct βx∗ of Eq. (2.43) under the conditions of Theorem 2.

From Eq. (2.3), the following holds for all k ≥ N :

|xh∗(k)− xs| ≤ βx(|xh∗(N)− xs|, k −N) (2.45)

30

Using Lemma 5 and taking into account that V rN,∞(x(n)) = V ∗N,∞(x(n)) (because

limNh→∞

λ(xh(Nh, x(N)) = λ(xs)), V∗N,∞(x(0)) is also upper-bounded by V ∗N,∞(x(n)) ≤

αv(|x(0)− xs|). Combining this with Assumption 1, the following can be obtained

αl(|xh∗(N)− xs|) ≤ l(xh∗(N), uh∗(N)) ≤ V ∗N,∞(x(n)) ≤ αv(|x(0)− xs|) (2.46)

Substituting the above into Eq. (2.45), the following holds for all k ≥ N :

|xh∗(k)− xs| ≤ βx(α−1l (αv(|x(0)− xs|)), k −N)

The right-hand-side of the above is a class KL function with respect to |x(0)−xs| and

k for k ≥ N . We can extend the above to account for k = 0, ..., N − 1 by noting that

|xh∗(k) − xs| ≤ α−1l (αv(|x(0) − xs|)) for all k = 0, ..., N − 1. βx∗ can be constructed

as follows

βx∗(|x(0)− xs|, k) =

maxα−1l (αv(|x(0)− xs|)), βx(α

−1l (αv(|x(0)− xs|)), 0)

k = 0, ..., N − 1.βx(α

−1l (αv(|x(0)− xs|)), k −N), k ≥ N

It can be verified that βx∗ defined above is a class KL function that satisfies Eq. (2.43).

Under the conditions of Theorem 3 , Eq. (2.44) can be proved following the similar

procedure. We provide a sketch of the proof. From Assumption 4, the following holds

for all k ≥ N

|xh∗(k)− xs| ≤ ask−N |xh∗(N)− xs| (2.47)

Based on Assumption 5 and Lemma 7, Eq. (2.46) can be further written as

c1|xh∗(N)− xs|p ≤ l(xh∗(N), uh∗(N)) ≤ V ∗N,∞(x(n)) ≤ av|x(0)− xs|p (2.48)

Combining Eqs. (2.47) and (2.48) it can be shown that Eq. (2.44) holds for all k ≥ N .

It can be also verified that Eq. (2.44) holds for all 0 ≤ k < N taking into account

that ask−N > 1 and that Eq. (2.48) holds if xh∗(N) is replaced with xh∗(k) for all

0 ≤ k < N .

Lemma 11 If the conditions of Theorem 2 are satisfied, then the following holds:

V ∗N,Nh(x(0)) ≤ V ∗N,∞(x(0)) + βλ(dmax, Nh) (2.49)

31

where βλ is defined in Lemma 4. Moreover, if the conditions of Theorem 3 are satis-

fied, then the following holds:

V ∗N,Nh(x(0)) ≤ V ∗N,∞(x(0)) + aλdmaxsNh (2.50)

where aλ is defined in Lemma 9.

Proof. First we prove Eq. (2.49) under the conditions of Theorem 2. Recall from

Eq. (2.21):

V ∗N,Nh(x(0)) ≤ V rN,Nh

(x(0)) + λ(xh(Nh, x∗r(N |0)))− λ(xs) (2.51)

Due to the positive-finiteness of the rotated cost,

V rN,Nh

(x(0)) ≤ V rN,∞(x(0)) = V ∗N,∞(x(0)) (2.52)

Using Lemma 4,

λ(xh(Nh, x∗r(N |0)))− λ(xs) ≤ βλ(|xh(Nh, x

∗r(N |0)))− xs|, k) ≤ βλ(dmax, Nh) (2.53)

Substituting Eqs. (2.52) and (2.53) into Eq. (2.51), Eq. (2.49) is obtained.

Following the similar arguments, Eq. (2.50) under the conditions of Theorem 3

can be obtained with Eq. (2.53) further written as:

λ(xh(Nh, x∗r(N |0)))− λ(xs) ≤ aλ|x∗r(N |0)− xs|sNh ≤ aλdmaxs

Nh (2.54)

because of Lemma 9. The details are omitted here for brevity.

Lemma 12 If Assumptions 1-3 hold, then there exists a class KL function βh∗ such

that the following holds for all x(0) ∈ XN :

∞∑k=K

l(xh∗(k), uh∗(k)) ≤ βh∗(|x(0)− xs|, K) (2.55)

Moreover, if Assumptions 1, 4, 5, 6 hold, then there exists a class KL function

βeh∗(|x(0) − xs|, K) which decreases to 0 exponentially fast as K increases such that

the following holds for all x(0) ∈ XN :

∞∑k=K

l(xh∗(k), uh∗(k)) ≤ βeh∗(|x(0)− xs|, K) (2.56)

32

Proof. First we prove Eq. (2.55) under Assumptions 1-3. To construct βh∗ , we first

note that there exists a class K∞ function αh∗ such that

∞∑k=K

l(xh∗(k), uh∗(k)) ≤ αh∗(|xh∗(K)− xs|) (2.57)

The above can be shown based on Lemma 1 by treating∞∑k=K

l(xh∗(k), uh∗(k)) as a

function of xh∗(K). Specifically, it can be verified that (i): if xh∗(K) = 0, then∞∑k=K

l(xh∗(k), uh∗(k)) = 0 because of the positive-definiteness of the rotated cost; (ii):

∞∑k=K

l(xh∗(k), uh∗(k)) is continuous at xh∗(K) = 0 because of the continuity Assump-

tion 2; and (iii):∞∑k=K

l(xh∗(k), uh∗(k)) is bounded because V ∗N,∞(x) =∞∑k=0


is bounded. Using Eq. (2.43) of Lemma 10 to replace xh∗(K) on the right-hand-side

of Eq. (2.57).∞∑k=K

l(xh∗(k), uh∗(k)) ≤ αh∗(βx∗(|x(0)− xs|, K))

Eq. (2.55) is thus established with βh∗ = αh∗(βx∗(|x(0)− xs|, K)).

Next we prove Eq. (2.56) under Assumptions 1, 4, 5, 6. It can be verified that the

following holds for all x(0) ∈ XN and K ≥ 0:

∞∑k=K

l(xh∗(k), uh∗(k)) ≤ av|xh∗(K)− xs|p (2.58)

where the positive constant av is defined in Eq. (2.25). The proof is similar to that of

Lemma 7 and is omitted here for brevity. Using Eq. (2.44) of Lemma 10 to replace

xh∗(K) on the right-hand-side of Eq. (2.58):

∞∑k=K

l(xh∗(k), uh∗(k)) ≤ av2ap

c1

s(k−N)p|x(0)− xs|p

It can be verified that the right-hand-side of the above is a classKL function βeh∗(|x(0)−

xs|, K) which decreases to 0 exponentially fast as K increases.

Now we are ready to state the main results of this subsection:

Theorem 6 If the conditions of Theorem 2 are satisfied, then there exist class KL

functions βK, βNh and βl such that the following holds:

JEMPCK ≤ Jh

∗

K + βK(|x(0)− xs|, K) + βNh(dmax, Nh) +Kβl(dmax, Nh) (2.59)

33

Moreover, if the conditions of Theorem 3 are satisfied, then the following holds

JEMPCK ≤ Jh

∗

K + βK(|x(0)− xs|, K) + βeNh(dmax, Nh) +Kβel (dmax, Nh) (2.60)

where βeNh and βel decreases to 0 exponentially fast as Nh increases.

Proof. First we prove Eq. (2.59) under the conditions of Theorem 2. Recall from

Eq. (2.41)

JEMPCK =

K−1∑n=0

l(x(n), u(n)) ≤ V ∗N,Nh(x(0))− V ∗N,Nh(x(K)) +Kβl(dmax, Nh)

Using Eq. (2.9) to replace the original objective function with the rotated objective

function:

JEMPCK ≤ V ∗N,Nh(x(0))− V ∗N,Nh(x(K))− λ(x(0)) + λ(x(K)) +Kβl(dmax, Nh) (2.61)

Based on Lemma 11,

V ∗N,Nh(x(0)) ≤ V ∗N,∞(x(0)) + βλ(dmax, Nh)

=K−1∑k=0

l(xh∗(k), uh∗(k)) +∞∑k=K

l(xh∗(k), uh∗(k)) + βλ(dmax, Nh)(2.62)

For the first term on the right-hand-side of Eq. (2.62):

K−1∑k=0

l(xh∗(k), uh∗(k)) = JhK + λ(x(0))− λ(xh∗(K)) (2.63)

Substituting Eqs. (2.63) and (2.55) of Lemma 12 into Eq. (2.62):

V ∗N,Nh(x(0)) ≤ JhK + λ(x(0))− λ(xh∗(K)) + βh∗(|x(0)− xs|, K) + βλ(dmax, Nh) (2.64)

From Lemma 4 and the positive-definiteness of the rotated cost, the following can be

obtained

V ∗N,Nh(x(K)) ≥ −βλ(dmax, Nh) (2.65)

Substituting Eqs. (2.64) and (2.65) into Eq. (2.61):

JEMPCK ≤ JhK+λ(x(K))−λ(xh∗(K))+βh∗(|x(0)−xs|, K)+2βλ(dmax, Nh)+Kβl(dmax, Nh)

(2.66)

34

Based on Lemma 1, there exists a class K∞ function αλ such that

|λ(x)− λ(xs)| ≤ αλ(|x− xs|) (2.67)

for all x ∈ XN . From Theorem 2, the following holds:

λ(x(K))− λ(xs) ≤ αλ(|x(K)− xs|) ≤ αλ(maxβn(|x(0)− xs|, K), βr(dmax, Nh))

which further yields:

λ(x(K))− λ(xs) ≤ αλ(βn(|x(0)− xs|, K)) + αλ(βr(dmax, Nh)) (2.68)

Using Lemma 10,

λ(xs)− λ(xh∗(K)) ≤ αλ(|xh∗(K)− xs|) ≤ αλ(βx∗(|x(0)− xs|, K)) (2.69)

Substituting Eqs. (2.68) and (2.69) into Eq. (2.66) yields:

JEMPCK ≤ Jh

∗K + αλ(βn(|x(0)− xs|, K)) + αλ(βr(dmax, Nh)) + αλ(βx∗(|x(0)− xs|, K))

+βh∗(|x(0)− xs|, K) + 2βλ(dmax, Nh) +Kβl(dmax, Nh)

Eq. (2.59) is thus proved with

βK(|x(0)−xs|, K) = αλ(βp(|x(0)−xs|, K))+αλ(βx∗(|x(0)−xs|, K))+βh∗(|x(0)−xs|, K)

βNh(dmax, Nh) = αλ(βr(dmax, Nh)) + 2βλ(dmax, Nh)

Under the conditions of Theorem 3, Eq. (2.60) can be proved following the similar

arguments with αλ(|x−xs|) in Eq. (2.67) replaced with Lλ|x−xs| where Lλ is the Lips-

chitz constant of λ on XN ; βr replaced with βer as in Theorem 3; βλ(dmax, Nh) replaced

with aλdmaxsNh because of Lemma 11. The resulting βeNh(dmax, Nh) is βeNh(dmax, Nh) =

Lλβer(dmax, Nh) + 2aλdmaxs

Nh . βl can be replaced with βel because of Theorem 5.

Remark 3 Comparing the results of Theorem 5 and Theorem 6, it can be seen that

the results of Theorem 6 are stronger. Theorem 5 can be recovered from Theorem 6 by

dividing K on both sides of Eq. (2.59) or Eq. (2.60) with K →∞ (the asymptotic per-

formance of h∗ is l(xs, us)). Note that the terms βK(|x(0)−xs|, K) and βNh(dmax, Nh)

or βeNh(dmax, Nh) do not affect the asymptotic performance. The vanishing of these

terms are due to the practical stability of the EMPC design as well as the stability of

35

the auxiliary controller h. The above results also call for the auxiliary controller to be

not only stabilizing but also to provide satisfying transient performance in the termi-

nal region. For strictly dissipative systems, stability implies asymptotic performance,

which is a weaker index than transient performance.

For the special case when exponential stability as in Theorem 4 is achieved, we have

the following results on the transient performance of the EMPC design:

Theorem 7 If the conditions of Theorem 4 are satisfied, then there exists a finite

N∗h such that the following holds for all Nh ≥ N∗h :

JEMPCK ≤ Jh

∗

K + βeK(|x(0)− xs|, n) + βeNh(|x(0)− xs|, Nh)

where βeK(|x(0) − xs|, n) decreases exponentially to 0 as n increases, βeNh(|x(0) −

xs|, Nh) decreases exponentially to 0 as Nh increases.

Proof. We prove Theorem 7 with N∗h defined in Eq. (2.37). From Eq. (2.36) and

Assumption 5, the following can be obtained:

V ∗N,Nh(x(n+ 1))− V ∗N,Nh(x(n))

≤(cλ(1 + sp)apsNhp

cVc1

2+ c2(1 + Lh

p)apsNhpcVc1

2− 1)l(x(n), u(n))

Let Le(Nh) = cλ(1 + sp)apsNhpcVc1

2+ c2(1 + Lh

p)apsNhpcVc1

2. It can be verified that Le

is a class L function of Nh that decreases to 0 exponentially fast with Le(N∗h) < 1.

Summing the above from n = 0 to K − 1 and taking into account that V ∗N,Nh(x(n))

is nonnegative because Theorem 4 is satisfied, the following can be obtained:

(1− Le(Nh))K−1∑n=0

l(x(n), u(n)) ≤ V ∗N,Nh(x(0))− V ∗N,Nh(x(K)) ≤ V ∗N,Nh(x(0)) (2.70)

From Theorem 4 and taking into account that Nh ≥ N∗h and Le(N∗h) < 1, the following

holds from Eq. (2.70):

K−1∑n=0

l(x(n), u(n)) ≤ cv|x(0)− xs|p

1− Le(N∗h)

Substituting the above back into Eq. (2.70), the following can be obtained:

K−1∑n=0

l(x(n), u(n)) ≤ V ∗N,Nh(x(0)) + Le(Nh)cv|x(0)− xs|p

1− Le(N∗h)(2.71)

36

From Lemma 7 and Assumption 5, and taking into account the positive-definiteness

of l, the following holds:

c1|x∗r(N |0)− xs|p ≤ l(x∗r(N |0), u∗r(N |0)) ≤ V rN,Nh

(x(0)) ≤ av|x(0)− xs|p

Sustituting the above into Eq. (2.54), Eq. (2.50) of Lemma 11 can be futher written

as

V ∗N,Nh(x(0)) ≤ V ∗N,∞(x(0)) + aλsNhavc1

|x(0)− xs|p

Sustituting the above into Eq. (2.71):

K−1∑n=0

l(x(n), u(n)) ≤ V ∗N,∞(x(0)) + aλsNhavc1

|x(0)− xs|p +Le(Nh)cv|x(0)− xs|p

1− Le(N∗h)(2.72)

V ∗N,∞(x(0)) =K−1∑k=0

l(xh∗(k), uh∗(k)) +∞∑k=K


Using Eq. (2.63) and Eq. (2.56) of Lemma 12 to replace the two parts on the right-

hand-side of the above

V ∗N,∞(x(0)) ≤ JhK + λ(x(0))− λ(xh∗(K)) + βeh∗(|x(0)− xs|, K) (2.73)

The left-hand-side of Eq. (2.71) can be written as:

K−1∑n=0

l(x(n), u(n)) = JEMPCK + λ(x(0))− λ(x(K)) (2.74)

Substituting Eqs (2.73) and (2.74) into Eq. (2.71):

JEMPCK ≤ JhK + λ(x(K))− λ(xh∗(K)) + βeh∗(|x(0)− xs|, K)

+aλsNhavc1

|x(0)− xs|p + Le(Nh)cv|x(0)− xs|p

1− Le(N∗h)

(2.75)

From Theorem 4, xs is exponentially stable under the proposed EMPC which im-

plies that there exists a class KL function βen(|x(0) − xs|, n) which decreases to 0

exponentially as n increases such that:

|x(n)− xs| ≤ βen(|x(0)− xs|, n)

Using the above and Eq. (2.44) of Lemma 10, and taking into account the Lipschitz

continuity of λ, the following holds:

λ(x(K))− λ(xh∗(K)) = λ(x(K))− λ(xs) + λ(xs)− λ(xh∗(K))

≤ Lλβen(|x(0)− xs|, K) + Lλβ

ex∗(|x(0)− xs|, K)

(2.76)

37

Substituting Eq. (2.76) into Eq. (2.75):

JEMPCK ≤ JhK + Lλβ

en(|x(0)− xs|, K) + Lλβ

ex∗(|x(0)− xs|, K) + βeh∗(|x(0)− xs|, K)

+aλsNhavc1


1− Le(N∗h)(2.77)

Taking

βeK(|x(0)−xs|, n) = Lλβen(|x(0)−xs|, K) +Lλβ

ex∗(|x(0)−xs|, K) +βeh∗(|x(0)−xs|, K)

and

βeNh(|x(0)− xs|, Nh) = aλsNhavc1


1− Le(N∗h)

Theorem 7 is proved.

Remark 4 The results of Theorems 6 and 7 indicate that for sufficiently large Nh,

the transient or asymptotic performance of the EMPC design is approximately upper-

bounded by the performance of the auxiliary controller h or its improved version h∗.

This implies that the EMPC either extends the feasibility region of the auxiliary con-

troller or improves its performance. Moreover, if h(x) is (near) optimal on the ter-

minal region Xf or around the optimal steady state xs, then the local optimality is

approximately preserved by the EMPC if a sufficiently large terminal horizon Nh is

used. Since our analysis is carried out for an arbitrary control horizon N , these results

also explain the computational efficiency of the EMPC design.

2.4 Numerical examples

Example 5.1 This example is a linearized continuously stirred tank reactor model

taken from [10, 28]. Consider the control system:

x(k + 1) =

(0.8353 00.1065 0.9418

)x(k) +

(0.00457−0.00457

)u(k) +

(0.55590.5033

)(2.78)

with the stage cost l(x, u) = |x|2 + 0.05u2. The state and input constraints are:

X = [−100, 100]2 and U = [−10, 10]. The optimal steady state that solves the steady-

state optimization problem of Eq. (2.2) is xs ≈ [3.5463, 14.6531]T and us ≈ 6.1637.

We choose the auxiliary controller to be the open-loop optimal steady-state input

38

0 50 100 15010

−4

10−3

10−2

10−1

100

101

k

|x(k

)−x s|

N=1

N=5

N=15

N=10Nh=5

Nh=10

Nh=20

Nh=15 N=20

Nh=1

Figure 2.1: Closed-loop system state trajectories under the proposed EMPC withN = 1 and Nh = 1, 5, 10, 15, 20 (solid lines), EMPC without terminal cost withN = 1, 5, 10, 15, 20 (dashed lines), and the auxiliary controller h (dash-dotted line).

h(x) = us with the terminal region Xf = x : |x − xs| ≤ 85. Under these set-

tings, it can be verified that all the conditions for Theorem 3 are satisfied. In par-

ticular, Assumption 1 is satisfied with a linear storage function λ(x) = cTx where

c = [−368.6684,−503.5415]T . The corresponding rotated stage cost is a quadratic

function that achieves its minimum at (xs, us).

In the simulations, we make comparisons between the following configurations:

(1) the proposed EMPC with control horizon N = 1 and different terminal horizons

Nh = 1, 5, 10, 15, 20; (2) An EMPC without terminal cost and with different control

horizons N = 1, 5, 10, 15, 20; (3) the auxiliary controller h(x) = us. The initial

state is x(0) = [4, 20]T .

Figure 2.1 shows the state trajectories under the different configurations. The

system state converges to the optimal steady state under the proposed EMPC design.

Moreover, the practical stability region shrinks exponentially fast as Nh increases.

These results agree with the results in Theorem 3. Figure 2.2 shows the transient

performance defined in Eq. (2.42) under different configurations from K = 50 to

150. The performance of the proposed EMPC design approaches that of the auxiliary

controller as Nh increases, agreeing with the results of Theorem 6. Note that the slope

39

50 100 1502900

3200

3500

K

J K N=5

N=10

N=1N

h=1

Nh=5

Nh=10

Figure 2.2: Transient performance under the proposed EMPC with N = 1 and Nh =1, 5, 10 (solid lines), EMPC without terminal cost with N = 1, 5, 10 (dashed lines), andthe auxiliary controller h (dash-dotted line).

Table 2.1: Shifted asymptotic performance (Jasy − l(xs, us)) and average optimizationevaluation time (CPU) of the proposed EMPC with N = 1

Nh = 1 Nh = 5 Nh = 10 Nh = 15 Nh = 20Jasy − l(xs, us) 1.3524 0.3149 0.0518 0.0086 0.0014

CPU(s) 0.0132 0.0160 0.0214 0.0234 0.0255

of JK decreases as K increases which corresponds to the vanishing term βK(|x(0) −

xs|, K) of Eq. (2.60).

The asymptotic performance and average optimization problem evaluation time

under the proposed EMPC and EMPC without terminal cost are shown in Table 2.1

and Table 2.2, respectively. The asymptotic performance converges to the optimal

steady state performance exponentially as Nh increases, which verifies our analysis

in Theorem 5. For roughly the same prediction horizon, the proposed EMPC design

achieves approximately the same level of convergence and performance as that of the

EMPC without terminal conditions, but with much less computational efforts. These

results demonstrate the computational efficiency of the proposed EMPC design.

40

Table 2.2: Shifted asymptotic performance (Jasy − l(xs, us)) and average optimizationevaluation time (CPU) of the EMPC without terminal cost.

N = 1 N = 5 N = 10 N = 15 N = 20Jasy − l(xs, us) 1.9581 0.4487 0.0720 0.0116 0.0018

CPU(s) 0.0120 0.0351 0.0709 0.1192 0.1921

Example 5.2 This example is also taken from [10]. Consider the control system:

x(k + 1) = 2x(k) + u(k) (2.79)

with state and input constraints: X = [−0.5, 0.5], U = [−2, 2]. The stage cost

to be minimized is l(x, u) = u2. Thus, the objective of the control problem is to

maintain the system state x inside the region X with the minimum control effort. It

can be easily checked that the optimal steady state is (xs, us) = (0, 0). The system

of Eq. (2.79) satisfies the strict dissipativity Assumption 1 with a quadratic storage

function λ(x) = −x2/2. The corresponding rotated stage cost of Eq. (2.6) is l(x, u) =

2u2 + 1.5x2 + 2xu, which is subject to polynomial bounds: x2 ≤ l(x, u) ≤ 3(x2 + u2).

Assumption 5 is thus satisfied with c1 = 1, c2 = 3 and p = 2. Assumption 7 is

satisfied with cλ = 0.5. We design the auxiliary controller h(x) = −1.5x, which

satisfies Assumption 4 with a = 1 and s = 0.5 on Xf = X. Assumption 6 is also

satisfied. Thus, all the conditions for Theorem 4 are satisfied.

In the simulations, we compare (1) the proposed EMPC with N = 1 and Nh =

1, 2, 5; (2) EMPC without terminal cost (but with terminal region constraint) with

N = 1, 5, 10, 15, 20; And (3) the auxiliary controller. The initial state is x(0) =

0.5. The state and stage cost under different scenarios are shown in Figure 2.3 and

Figure 2.4 , respectively. The transient performance JK with K = 30 under different

scenarios and the corresponding average optimization evaluation times are shown

in Table 2.3 and Table 2.4. From Figure 2.3, it can be seen that EMPC with the

proposed terminal cost exponentially stabilizes the optimal steady state (note that

the y-axis is in logarithmic coordinate), which agrees with Theorem 4. Figure 2.3

and Table 2.4 verify our analysis in Theorem 7 that the transient performance of

the proposed EMPC design approaches that of the auxiliary controller exponentially

fast as Nh increases. In this example, employing the extended terminal horizon not

41

Table 2.3: Transient performance (J30) and optimization evaluation time (CPU) of theproposed EMPC with terminal cost with N = 1.

Nh = 1 Nh = 2 Nh = 5J30 0.7714 0.7508 0.7500

CPU(s) 0.0208 0.0210 0.0215

only improves the computational efficiency for EMPC, but also leads to exponential

stability which cannot be achieved by EMPC without terminal condition.

0 5 10 15 20 25 3010

−10

10−8

10−6

10−4

10−2

100

k

x(k

)

N=10

N=5

N=1

N=15

N=20N

h=1

Nh=2

Nh=5

Figure 2.3: State trajectories of the closed-loop system under the proposed EMPCwith N = 1 and Nh = 1, 2, 5 (solid lines), EMPC without terminal cost with N =1, 5, 10, 15, 20 (dashed lines), and the auxiliary controller h (dash-dotted line).

2.5 Summary

In this chapter, we provided a general framework to analyze the stability and perfor-

mance of EMPC with extended terminal horizon. While a finite terminal horizon is

in general not sufficient to ensure stability of the optimal steady state, it is sufficient

to achieve practical stability for strictly dissipative systems under mild assumptions.

Further conditions to ensure the exponential shrinkage of the practical stability region

are provided. For a special case including conventional MPC with positive-definite

42

0 5 10 15 20 25 3010

−20

10−15

10−10

10−5

100

k

l(x,u

)

N=15

N=20

N=10

N=5

N=1

Nh=1

Nh=5

Nh=2

Figure 2.4: Stage costs of the closed-loop system under the proposed EMPC with N = 1and Nh = 1, 2, 5 (solid lines), EMPC without terminal cost with N = 1, 5, 10, 15, 20(dashed lines), and the auxiliary controller h (dash-dotted line).

Table 2.4: Transient performance (J30) and optimization evaluation time (CPU) of theEMPC without terminal cost.

N = 1 N = 5 N = 10 N = 15 N = 20J30 7.500 0.8088 0.7501 0.7500 0.7500

CPU(s) 0.0212 0.0356 0.0460 0.0712 0.0899

43

stage costs, exponential stability can be achieved. Performance of the EMPC is also

shown to be approximately upper-bounded by that of the auxiliary controller if a large

terminal horizon is used. These results provide insights into the intrinsic properties

of EMPC and also explain the computational efficiency of the EMPC design.

44

Chapter 3

Economic MPC with localoptimality

In this chapter, we design a terminal cost which preserves the local optimality for

EMPC. We first show, based on the strong duality and second order sufficient con-

dition (SOSC) of the steady-state optimization problem, that the optimal operation

of the system is locally equivalent to an LQR controller. The proposed terminal cost

is constructed as the value function of the LQR controller plus a linear term charac-

terized by the Lagrange multiplier associated with the steady-state constraint. From

the perspective of dissipative systems, the linear term corresponds to a linear storage

function for the system. EMPC with the proposed terminal cost is stabilizing with an

appropriately chosen control horizon, and preserves the local optimality of the LQR

controller. Simulation results of an isothermal CSTR verify our analysis.

3.1 Preliminaries

3.1.1 Notation

Throughout this chapter, the operator | · | denotes the Euclidean norm of a scalar or

a vector. The notation int(X) denotes the interior of the set X. The symbol Br(xs)

denotes the open ball centered at xs with radius r such that Br(xs) := x : |x−xs| <

r. The operator λmin(max)(H) denotes the minimum (maximum) eigenvalue of matrix

H. The symbol ‘\’ denotes set substraction such that A \ B := x ∈ A, x /∈ B.

45


We consider the nonlinear discrete state-space model:

x(k + 1) = f(x(k), u(k)) (3.1)

where x ∈ Rnx and u ∈ Rnu denote the system state and input respectively, k ≥ 0 is

the discrete time index. The performance of the system is measured by an economic

cost function l(x, u) : Rnx × Rnu → R. It is assumed that both f(x, u) and l(x, u)

are twice continuously differentiable. The system state and input are subject to

constraint g(x) ≤ 0 and h(u) ≤ 0 respectively, where the sets X := x : g(x) ≤ 0

and U := u : h(u) ≤ 0 are compact. For simplicity of discussion, we assume that

the set X is forward-invariant. That is, for any x ∈ X, there exists u ∈ U such that

f(x, u) ∈ X. The forward invariance of the constraint set ensures recursive feasibility

of the moving horizon optimization problems. This condition can be relaxed, see e.g.,

[6] and [9] (Sections 8.2-8.3) for results in the context of MPC with positive definite

stage cost.

We use (xs, us) to denote the optimal steady state and input of the system, which

is the solution to the following steady-state optimization problem:

(xs, us) = arg minx,u

l(x, u)

s.t. x = f(x, u)

g(x) ≤ 0

h(u) ≤ 0

(3.2)

Note that the pair (xs, us) is not necessarily a minimizer of the economic cost function

l(x, u).

3.2 Optimal operation of the system

A key difference between conventional MPC with positive-definite stage cost and

economic MPC is that in EMPC, steady-state operation is not necessarily the optimal

operation. For generic nonlinear systems, it is possible that a dynamic trajectory (e.g.,

a periodic orbit) outperforms the average performance of steady-state operation. The

optimality of steady-state operation is shown to be closely connected to the notion of

46

dissipativity or the so-called turnpike property [25], [26], [27]. Verifying dissipativity

of the system amounts to finding a storage function, which is in general a difficult

task. In this chapter, we consider systems satisfying the strong duality condition (see

e.g., [38] Section 5) where a linear storage function can be found. Similar approach

was used in [28] to establish stability of EMPC with point-wise terminal constraint.

We make the following assumption on the system:

Assumption 8 The steady-state optimization problem of Eq. (3.2) satisfies strong

duality and second order sufficient condition (SOSC). Moreover, the state and input

constraints are not active at the optimal steady state (xs, us). That is, g(xs) < 0,

h(us) < 0.

Lemma 13 If Assumption 8 holds, then there exists a positive scalar cl > 0 such

that the following holds for all x ∈ X and u ∈ U:

l(x, u) + λTs (f(x, u)− x)− l(xs, us) ≥ cl(|x− xs|2 + |u− us|2) (3.3)

where λs is the KKT multiplier associated with the equality constraint f(x, u)−x = 0.

Proof. Consider the Lagrangian:

L(w, λs, µs, νs) = l(x, u) + λTs (f(x, u)− x) + µTs g(x) + νTs h(x) (3.4)

where w = [xT , uT ]T with ws = [xTs , uTs ]T , λs, µs and νs are the KKT multipliers.

Since strong duality holds, L(ws, λs, µs, νs) = l(xs, us). Also taking into account the

SOSC: ∂L∂w|w=ws = 0, and H = ∂2L

∂w2 |w=ws > 0, the Taylor series of L(w, λs, µs, νs) at

w = ws can be written as follows:

L(w, λs, µs, νs) = l(xs, us) + (w − ws)TH(w − ws) +O(|w|2)

Due to the positive definiteness of H, for any positive scalar cs such that 0 < cs <

λmin(H), there exists ε > 0 such that the following holds for all w ∈ Bε(ws):

L(w, λs, µs, νs)− l(xs, us) ≥ cs(|x− xs|2 + |u− us|2)

Substituting Eq. (3.4) into the above and taking into account that λs = 0 and νs = 0

(because of the strict complimentary condition and the assumption that g(xs) < 0,

h(us) < 0), the following holds for all w ∈ Bε(ws):

l(x, u) + λTs (f(x, u)− x)− l(xs, us) ≥ cs(|x− xs|2 + |u− us|2)

47

The quadratic lower bound on Bε(ws) can be extended to the compact set Z := w :

x ∈ X, u ∈ U. Specifically, let lmax = supl(x, u)+λTs (f(x, u)−x)−l(xs, us) : w ∈ Z

and lmin = inf|x−xs|2 + |u−us|2 : w ∈ Z \Bε(ws), then cl = maxcs, lmax

lmin satisfies

Eq. (3.3) for all x ∈ X and u ∈ U.

Assumption 8 is slightly stronger than the assumptions made in [28]. In [28],

the left-hand-side of Eq. (3.3) (the rotated cost) is lower bounded by a K∞ function

of |x − xs|. In this chapter, a quadratic lower bound with respect to (xs, us) is

assumed. This will allow us to find the locally optimal controller and to approximate

the corresponding economic cost using linear quadratic regulator (LQR) approach.

Remark 5 The strong duality condition is satisfied for convex optimization problems

with affine equality constraints satisfying the Slater condition [38]. In the context

of control systems, it is satisfied for the steady-state optimization of linear systems

with convex cost functions and constraints (also satisfying the Slater condition). It

is important to note that Assumption 8 is rather strong for systems with optimal

steady-state operation. A more general condition which implies optimal steady-state

operation is dissipativity [2]. The merit of Assumption 8 is that it can be verified

numerically and provides us with a linear storage function that can be used to design

the terminal cost. Specifically, once (xs, us) and the corresponding KKT multipliers

are obtained, one can verify Eq. (3.3) by carrying out the following optimization:

minx,u

l(x, u) + λTs (f(x, u)− x)− l(xs, us)

s.t. g(x) < 0

h(u) < 0

If (xs, us) is the optimal solution to the above optimization problem, and if the hessian

matrix of the objective function is positive definite at (xs, us), then Assumption 8

holds.

Under Assumption 8, the system is strictly dissipative with respect to the stage

cost l(x, u)−l(xs, us), with a linear storage function −λsx for all x ∈ X, u ∈ U. Under

further controllability conditions (which we will make in Assumption 9), Assumption 8

implies that the system is suboptimally operated off steady state [25], [26], [27].

Specifically, this means that the asymptotic performance, or the infinite-time average

48

performance, is lower bounded by the steady-state economic performance:

limN→∞

1

N

N∑k=0

l(x(k), u(k)) ≥ l(xs, us)

with the equality attained only if limk→∞

x(k) = xs. The asymptotic performance how-

ever, is a relatively weak performance index which does not reflect transient perfor-

mance of the system. To this end, we consider the following infinite-time transient

performance:

J∞(x) = inf∞∑k=0


)(3.5a)

s.t. x(k + 1) = f(x(k), u(k)) (3.5b)

g(x(k)) ≤ 0 (3.5c)

h(u(k)) ≤ 0 (3.5d)

x(0) = x (3.5e)

limk→∞

x(k) = xs (3.5f)

The goal of EMPC is to approximate the above infinite-horizon optimization problem

in a receding horizon fashion. We make the following controllability assumption:

Assumption 9 J∞(x) <∞ for all x ∈ X. Moreover, the linearized system at (xs, us)

with [A,B] = [∂f∂x|(xs,us), ∂f∂u |(xs,us)] is controllable.

The assumption that J∞(x) <∞ for all x ∈ X is equivalent to assuming stabiliz-

ability of the system on (x, u) : x ∈ X, u ∈ U and that the system can be driven to

the steady state with a finite cost. Note that J∞(x) > −∞ holds automatically due

to strong duality. Now let us investigate the property of J∞(x) for x near xs.

Lemma 14 If Assumption 8 and Assumption 9 hold, then there exists ε > 0 such

that for all x ∈ Bε(xs), J∞(x) can be locally approximated as:

J∞(x) = λs(x− xs) +1

2(x− xs)TP (x− xs) +O(|x− xs|2)

where P is the solution to the following discrete time algebraic Riccati equation:

P = ATPA− (ATPB +N)(R +BTPB)−1(BTPA+NT ) +Q (3.6)

49

where

H =∂2L

∂w2|w=ws =

[ Q NNT R

]> 0

is the Hessian matrix of the Lagrangian evaluated at (xs, us). The corresponding

locally optimal controller is

u = KLQ(x− xs) + us (3.7)

where KLQ = (R +BTPB)−1BTPA.

Proof. Let us define the rotated stage cost

l(x, u) = l(x, u) + λTs (f(x, u)− x)− l(xs, us) (3.8)

which is positive definite with respect to (xs, us) on x ∈ X, u ∈ U, as established

in Lemma 13. From the definition of l(x, u) and taking into account the constraint

of Eq. (3.5f), the objective function of Eq. (3.5a) can be rewritten in the following

rotated form: ∑∞k=0


)=

∑∞k=0 l(x(k), u(k))− λsx(0) + λs lim

k→∞x(k)

=∑∞

k=0 l(x(k), u(k))− λs(x− xs)Consider the following optimization problem:

J∞ = inf∑∞

k=0 l(x(k), u(k))

s.t. (3.5b)− (3.5e)(3.9)

The solutions to the optimizations of Eq. (3.5) and Eq. (3.9) are identical, because the

objective functions of the two problems differ by a bounded constant term λs(x−xs),

and the constraint sets are the same except for Eq. (3.5f) which is automatically

satisfied in Eq. (3.9) because of Assumption 9 and the positive-definiteness of l(x, u).

Eq. (3.9) is essentially a conventional MPC with positive-definite stage cost which is

quadratically lower bounded. It can be shown by geometric methods (see e.g., [39] [7]

for results of continuous systems which can be readily extended to discrete systems)

that there exists ε > 0 such that for all x ∈ Bε(xs), the optimal trajectory of Eq. (3.9)

does not violate the state and input constraints, and that J∞(x) can be approximated

by:

J∞ =1

2(x− xs)TP (x− xs) +O(|x− xs|2)

50

where P is defined in Eq. (3.6). Thus, for all x ∈ Bε(xs):

J∞ = J∞ + λs(x− xs)= λs(x− xs) + (x− xs)TP (x− xs) +O(|x− xs|2)

This proves Lemma 14.

Lemma 14 shows that the control problem of Eq. (3.5) can be locally approximated

as a linear quadratic regulator design problem. The optimal objective function can be

locally approximated as the corresponding infinite-horizon LQR cost plus the differ-

ence of the storage function at the steady state and the initial state. In the following,

we will use this approximation as the terminal cost for EMPC and discuss the stability

and performance of the EMPC design.

3.3 EMPC with the proposed terminal cost

At a time instant n, our EMPC design is formulated as the following optimization

problem:

minu(0),...,u(N−1)

N−1∑k=0

l(x(k), u(k)) + Vf (x(N)) (3.10a)

s.t. x(k + 1) = f(x(k), u(k)), k = 0, ..., N − 1 (3.10b)

x(0) = x(n) (3.10c)

g(x(k)) ≤ 0, k = 0, ..., N − 1 (3.10d)

h(u(k)) ≤ 0, k = 0, ..., N − 1 (3.10e)

where x(k) denotes the predicted state trajectory, x(n) is the state measurement at

time instant n. The terminal cost Vf (x) is the local second order approximation of

J∞(x) characterized in Lemma 14. That is,

Vf (x) = λs(x− xs) +1

2(x− xs)TP (x− xs)

The optimal solution to the above optimization problem is denoted as u∗(k|n), k =

0, ..., N − 1. The corresponding optimal state trajectory is x∗(k|n), k = 0, ..., N . The

manipulated input of the closed-loop system under the EMPC at a time instant n is:

u(n) = u∗(0|n). At the next sampling time n + 1, the optimization of Eq. (3.10) is

re-evaluated.

51

In the following, we consider the stability and performance of the EMPC design

of Eq. (3.10). To proceed, let us consider the following MPC design

minu(0),...,u(N−1)

∑N−1k=0 l(x(k), u(k)) + Vf (x(N))

s.t. (3.10b)− (3.10e)(3.11)

where Vf (x) = 12(x−xs)TP (x−xs), l(x, u) is the rotated stage cost defined in Eq. (3.8).

It can be easily checked that the objective functions of Eq. (3.10) and Eq. (3.11) only

differ by a constant term λsx(0) = λsx(n). Thus, the two MPCs deliver the same

solution. Proving stability of the proposed EMPC design is equivalent to proving

stability of the MPC of Eq. (3.11), which has positive definite stage cost and terminal

cost. This task has been dealt with in [7] and [8] where the discussions are made

under more general settings. In the following, we provide a tailored proof for the

task at hand. Let us use V ∗N(x) to denote the optimal objective function value of

Eq. (3.11) with control horizon N .

Lemma 15 There exists a positive scalar cV∞ such that the following holds for all

x ∈ X:

V ∗∞(x) ≤ cV∞|x− xs|2 (3.12)

Proof. From the proof of Lemma 14, Vf (x(N)) is a local quadratic approximation

of V ∗∞(x) (note that limN→∞

x(N) = xs and the terminal cost Vf (x(N)) can be dropped

for N → ∞). Thus, for any positive scalar cs > λmax(P ), there exists ε > 0 such

that the following holds for all x ∈ Bε(xs): V ∗∞(x) ≤ cs|x−xs|2. Following the similar

arguments as in the proof of Lemma 13, and based on the controllability assumption

that V ∗∞(x) is bounded on X, the quadratic upper bound of V ∗∞(x) can be extended

from the local region Bε(xs) to X. That is, there exists a positive scalar cV∞ such that

Eq. (3.12) holds for all x ∈ X.

Lemma 15 shows that the infinite-horizon cost of the MPC design of Eq. (3.11)

is quadratically upper bounded. Our final proof requires a stronger condition (

Lemma 16) that V ∗N(x) is quadratically upper bounded for all N ≥ 1 and x ∈ X.

This condition is stronger than Lemma 15 because V ∗N(x) is not necessarily an in-

creasing sequence with respect to N .

52

Lemma 16 There exists a positive scalar cVf such that the following holds for all

x ∈ X and N ≥ 1:

V ∗N(x) ≤ cVf |x− xs|2 (3.13)

Proof. Let us use x∞(k, x) and u∞(k, x), k ≥ 0 to denote the state and input

trajectories of V ∗∞(x) respectively. The following holds:

V ∗N(x) ≤N−1∑k=0

l(x∞(k, x), u∞(k, x)) + Vf (x∞(N, x)) (3.14)

Since l is positive definite, and from Lemma 15:

N−1∑k=0

l(x∞(k, x), u∞(k, x)) ≤ V ∗∞(x) ≤ cV∞|x− xs|2 (3.15)

So we only need to establish a quadratic upper bound of Vf (x∞(N, x)). From Lemma 13

and Lemma 15:

cl|x∞(N, x)− xs|2 ≤ l((x∞(N, x), u∞(N, x))) ≤ cV∞|x− xs|2

Thus, Vf (x∞(N, x)) = |x∞(N, x)− xs|2P≤ λmax(P )|x∞(N, x)− xs|2

≤ λmax(P )cV∞cl|x− xs|2

(3.16)

Substituting Eqs. (3.15) and (3.16) into Eq. (3.14), Lemma 16 is proved with cVf =

cV∞ + λmax(P )cV∞cl

Theorem 8 Consider the EMPC of Eq. (3.10) with x(0) ∈ X. If Assumption 8 and

Assumption 9 hold, then there exists a finite N∗ such that the EMPC is stabilizing

for all N ≥ N∗.

Proof. Let us use xN(k, x), k = 0, ..., N and uN(k, x), k = 0, ..., N − 1 to denote

the state and input trajectories of V ∗N(x) respectively. From Lemma 16, V ∗N(x) =N−1∑k=0

l(xN(k, x), uN(k, x))+ Vf (xN(N, x)) ≤ cVf |x−xs|2. Since l and Vf are all positive

definite, the above implies that there exists a p ∈ 1, ..., N − 1 (we assume N ≥ 2)

such that l(xN(p, x), uN(p, x)) ≤cVf

N − 1|x − xs|2. Using Lemma 13, this further

implies that

|xN(p, x)− xs|2 ≤cVf

(N − 1)cl|x− xs|2 (3.17)

53

Now let us consider V ∗N(xN(1, x)), which is the optimal value of the subsequent opti-

mization problem. A candidate solution to the subsequent optimization is to adopt

the state and input trajectories of xN(k, x) and uN(k, x) for k = 1, ...p− 1, and then

evaluate V ∗N−p(xN(p, x)) to fill up the rest state and input trajectories. The objective

function of this candidate solution, which is no smaller than V ∗N(xN(1, x)), is:

V ∗N(xN(1, x)) ≤p−1∑k=1

l(xN(k, x), uN(k, x)) + V ∗N−p(xN(p, x)) (3.18)

From Lemma 13 and the positive definiteness of the rotated stage and terminal cost,

the first term of the right-hand-side of Eq. (3.18) is bounded by:p−1∑k=1

l(xN(k, x), uN(k, x)) ≤ V ∗N(x)− l(x, uN(0, x))

≤ V ∗N(x)− cl|x− xs|2(3.19)

From Lemma 16 and Eq. (3.17), the second term of the right-hand-side of Eq. (3.18)

is bounded by:

V ∗N−p(xN(p, x)) ≤cVf

2

(N − 1)cl|x− xs|2 (3.20)

Substituting Eqs. (3.19) and (3.20) into Eq. (3.18), we have:

V ∗N(xN(1, x)) ≤ V ∗N(x) + (cVf

2

(N − 1)cl− cl)|x− xs|2

Thus, it is clear that the value function V ∗N(x) is a Lyapunov function of the closed-

loop system ifcVf

2

(N−1)cl− cl < 0 which is satisfied if N > (

cVfcl

)2 + 1. This proves

Theorem 13.

Theorem 13 shows that EMPC with the proposed terminal cost is stabilizing if

a sufficiently large N is used. The minimum value of N required to ensure stability

could be much smaller than what can be inferred from the above proof. For more

discussions on the size of N to ensure stability, interested readers may refer to [9],

Chapter 6.

Theorem 9 The EMPC of Eq. (3.10) is locally equivalent to the infinite-horizon

LQR controller of Eq. (3.7)

Proof. Let x(k) = x(k)− xs and u(k) = u(k)− us, consider the following quadratic

programming (QP):

min∑N−1

k=0 x(k)THx(k) + x(N)TPx(N)

s.t. x(k + 1) = Ax(k) +Bu(k)(3.21)

54

where H, P , A and B are defined in Lemma 14. The above QP delivers the LQR

control law characterized in Eq. (3.7). If the system state is sufficiently close to xs

such that no state or input constraints are active, the MPC of Eq. (3.11) which is

an equivalent transformation of our EMPC design, is locally equivalent to the QP of

Eq. (3.21). Thus the two optimization problems have locally first order equivalent

solutions, as a result of Theorem 16 in [40].

3.4 Case study

This example is taken from [2], [28]. Consider a single first-order irreversible chemical

reaction in an isothermal CSTR: A → B, where the reactant A is converted into

product B. The material balances of the process are:

dcAdt

=Q

VR(cAf − cA)− krcA

dcBdt

=Q

VR(cBf − cB) + krcA

(3.22)

where cA and cB are the molar concentrations of A and B respectively, cAf and cBf

are feed concentrations of A and B, Q is the volumetric flow through the reactor,

VR is the volume of the reactor, kr is the reaction rate constant. The corresponding

process parameters are shown in Table 3.1

Table 3.1: Process parameters

cAf cBf VR kr1 mol/L 0 L 10 L 1.2 L/(mol min)

In the controller design, the flow rate 0 ≤ Q ≤ 20L/min is the manipulated control

input. The economic objective is le(cA, xB, Q) = −2QcB + 0.5Q, which corresponds

to maximizing the product B and meanwhile minimizing the operational cost. The

optimal steady state of the system is cA = cB = 0.5mol/L, Q = 12L/min. Quadratic

penalty terms are added to the economic cost function to make the steady-state

optimization strongly dual: l(cA, xB, Q) = −2QcB + 0.5Q + |cA − 0.5|2QA + |cB −

0.5|2QB + |Q − 12|2R, where QA = QB = R = 0.505. The Lagrangian of the the

55

steady-state optimization with the regularized cost l(cA, xB, Q) is: L(cA, xB, Q) =

l(cA, xB, Q) + λTf(cA, xB, Q) + [−Q, Q − 20], where f(cA, xB, Q) denotes the right-

hand-side of Eq. (3.22). The optimal KKT multipliers are λ = [−10,−20]T and

ν = [0, 0]T . The corresponding Hessian matrix is:

H =

[ 1.01 0 10 1.01 01 0 1.01

]

which is positive definite with eigenvalues 0.01, 1.01 and 2.01. Thus, Assumption 8

is satisfied. The sampling time is 0.2 minutes. The discrete linearized model at the

optimal steady state is

A =[ 0.6188 0

0.1678 0.7866

]B =

[ 0.0079−0.0079

]Our proposed terminal cost for EMPC is

Vf (x) = λT (x− xs) +1

2(x− xs)TP (x− xs)

where

P =[ 0.4036 0.7048

0.7048 2.6490

]In the simulation, we compare three EMPC configurations: (1) EMPC with the pro-

posed terminal cost, (2) EMPC without terminal cost, and (3) EMPC with terminal

cost for stability [2] where the candidate terminal cost is

λT (x− xs) +1

2(x− xs)TPs(x− xs)

with Ps =[ 117.1 34.1

34.1 132.5

]. The initial state is [cA, cB] = [0.3, 2.2]. For comparison

purposes, we use a short horizon N = 2. The closed-loop system states, inputs and

stage costs are shown in Fig. 4.2, Fig. 4.3 and Fig. 3.3, respectively. The steady states

of the two EMPCs with terminal cost are the optimal steady state xs = [0.5, 0.5]T

and us = 12, whereas the steady state of EMPC without terminal cost is slightly off

the optimal with xs = [0.5049, 0.4961]T and us = 12.2371. The overall performance

J =∑l(x, u)− l(xs, us) of each configurations shown in Fig. 3.3 are summarized in

Table 3.2. It is seen from these results that EMPC with terminal cost for stability [2]

yields the fastest convergence but with relatively poor performance. EMPC without

56

Table 3.2: Process parameters

Proposed No terminal cost Terminal cost for stabilityJ -177.94 -174.13 -142.20

0 1 2 3 4 5 6

Time (h)

0.5

1

1.5

2

x (m

ol /

L)

cB

cA

Figure 3.1: Closed-loop state trajectories under EMPC with the proposed terminal cost(solid lines), EMPC without terminal cost (dashed lines), and EMPC with terminal costfor stability [2] (dash-dotted line).

terminal cost provides near-optimal transient performance but with the steady state

slightly off the optimal (due to the indefinite cost). EMPC with the proposed terminal

cost achieves both the near-optimal transient performance and stability of the optimal

steady state.

3.5 Summary

In this chapter, we propose a terminal cost for EMPC which preserves local optimality.

The EMPC design provides a balanced solution for EMPC without terminal condition

and EMPC with terminal cost for stability. Our results are derived based on the

strong duality and second order sufficient condition of the steady-state optimization

problem. Our future work will consider more general conditions under which suitable

57

0 1 2 3 4 5 6

Time (h)

12

14

16

18

20

Q (

L/m

in)

Figure 3.2: Closed-loop input trajectories under EMPC with the proposed terminal cost(solid lines), EMPC without terminal cost (dashed lines), and EMPC with terminal costfor stability [2] (dash-dotted line).

0 1 2 3 4 5 6

Time (h)

-45

-30

-15

0

l(x,u

)-l(x

s,us)

Figure 3.3: Closed-loop stage costs under EMPC with the proposed terminal cost (solidlines), EMPC without terminal cost (dashed lines), and EMPC with terminal cost forstability [2] (dash-dotted line).

58

storage functions of dissipative systems can be constructed and employed for terminal

cost design.

59

Chapter 4

Economic MPC for scheduledswitching operations

Switched systems consist of a set of subsystems or operating modes and a switching

schedule that specifies the switching between each operating modes. In the operation

of chemical processes, switching operations may arise under various situations. For

instance, a change in the raw material, the hybrid nature of the system stemming

from the discrete components such as switches, or the switching between different

controllers necessitated by different performance criteria. Stability and design of

switched systems have been a heated research area in the past decades (e.g., [41,

42, 43, 44]). Problems of interest include the controller design and stability analysis

for switched systems under arbitrary switching schedule, or identifying a class of

stabilizing switching schedules.

In this chapter, we extend the EMPC design in Chapter 2 for scheduled switching

operations. In the proposed approach, EMPC operations are divided into two phases.

If the current time is far away from the next scheduled mode switching time, EMPC

takes infinite-time operation under the current operating mode. If the scheduled mode

switching time is within the prediction horizon of the infinite-time EMPC design,

EMPC is operated in a mode transition phase. The proposed EMPC design enjoys

enlarged feasibility regions and practically improved performance than the auxiliary

controllers. Sufficient conditions to ensure recursive feasibility of the proposed EMPC

design are established. The simulation example of a chemical process demonstrates

the applicability and effectiveness of our approach.

60

4.1 Preliminaries

4.1.1 Notation

Throughout this chapter, the operator | · | denotes the Euclidean norm of a scalar or

a vector. The notation int(X) denotes the interior of the set X. The symbol Br(xs)

denotes the open ball centered at xs with radius r such that Br(xs) := x : |x−xs| <

r. A continuous function α : [0, a) → [0,∞) is said to belong to class K if it is

strictly increasing and satisfies α(0) = 0. A continuous function σ : [0,∞) → [0, a)

is said to belong to class L if it is strictly decreasing and satisfies limx→∞

σ(x) = 0. A

continuous function β : [0, a) × [0,∞) → [0,∞) is said to belong to class KL if for

each fixed r, β(r, s) belongs to class L, and for each fixed s, β(r, s) belongs to class

K.


We consider a class of switched nonlinear systems composed of p operating modes

described by the following discrete state-space model:

x(k + 1) = fM(k)(x(k), u(k)) (4.1)

where x ∈ Rnx denotes the state vector and u ∈ Rnu denotes the input vector.

The function M(k) : Z≥0 → I characterizes the prescribed switching schedule with

I := 1, 2, ..., p. We will use minr and mout

r to denote the time instant when, for the

r-th time, the system of Eq. (4.1) is switched in and out of mode m ∈ I, respectively.

Specifically, for k ∈ [minr ,m

outr ], M(k) = m and the system (4.1) is represented by

x(k + 1) = fm(x(k), u(k)).

Under an operating mode m ∈ I, the system state and input are subject to

constraint x ∈ Xm and u ∈ Um where Xm ⊂ Rnx and Um ⊂ Rnu are compact sets. It

is assumed that under each operating mode m, there exists an optimal steady state

(xssm, ussm) that uniquely solves the following steady-state optimization problem:

(xssm, ussm) = arg min

x,ulm(x, u)

s.t. x = fm(x, u)x ∈ Xm

u ∈ Um

(4.2)

61

where lm(x, u) : Rnx × Rnu → R is the economic stage cost function under the

operating mode m.

4.1.3 A set of auxiliary controllers

It is assumed that under each operating mode m ∈ I, there exists an explicit controller

u = hm(x) which renders the optimal steady state xssm asymptotically stable with

ussm = hm(xssm) while satisfying the state and input constraints for all x ∈ Dm, where

Dm ⊆ Xm is a compact set containing xssm in its interior. In the remainder, the region

Dm will be referred to as the stability region of the controller hm(x) under mode m.

We use xhm(k, x) to denote the closed-loop state trajectory under the controller hm

at time instant k with the initial state xhm(0, x) = x. The above assumptions imply

that there exists a class KL function βxm such that:

|xhm(k, x)− xssm| ≤ βxm(|x− xssm|, k)xhm(k, x) ∈ Dm

hm(xhm(k, x)) ∈ Um

(4.3)

for all k ≥ 0 and x ∈ Dm. In addition, we assume that the optimal steady state of

an operating mode m, xssm, lies in the interior of the stability region of the auxiliary

controller in the subsequent operating mode. This assumption is made explicit as

follows:

Assumption 10 For any two consecutive operating modes m, l ∈ I as prescribed by

M(k) such that moutr + 1 = linq , the following holds: xssm ∈ int(Dl).

Assumption 10 allows us to conclude the following result:

Proposition 1 If Assumption 10 holds, then there exists a finite Nml such that

xhm(k, x) ∈ Dl for all k ≥ Nml and x ∈ Dm.

Proof Since xssm ∈ int(Dl), there exists ε > 0 such that Bε(xssm) ⊂ Dl. Taking into

account the asymptotic stability of the controller hm(x) characterized in Eq. (4.3),

there exists a finite Nml such that βxm(|x− xssm|, k) ≤ ε for all k ≥ Nml and x ∈ Dm,

which implies that xhm(k, x) ∈ Dl for all k ≥ Nml and x ∈ Dm.

Proposition 1 shows that there is always a finite time in which the system state

under the operating mode m and controller hm(x) will be driven into the stability

62

region of its subsequent operating mode l. Let us define the minimal time interval

satisfying Proposition 1 as follows:

N∗ml := minNml : xhm(k, x) ∈ Dl, ∀k ≥ Nml ≥ 0,∀x ∈ Dm (4.4)

Note that the minimal time intervalN∗ml essentially depends on the convergence rate of

the controller hm(x) as well as the overlapping of the stability regions Dm and Dl. For

instance, if Dm = Dl, then N∗ml = 0. Based on N∗ml, we make the following assumption

on the lengths of the operating modes according to the prescribed schedule M(k):

Assumption 11 For any two consecutive operating modes m, l ∈ I as prescribed by

M(k) such that moutr + 1 = linq , the length of the operating mode m is no less than

N∗ml. That is, moutr −min

r ≥ N∗ml, where N∗ml is defined in Eq. (4.4).

Under Assumption 10 and Assumption 11, the set of controllers hm(x), m ∈ I

forms a feasible set of controllers for the scheduled switching operations:

Proposition 2 Consider the system of Eq. (4.1) in closed-loop under the scheduled

switching operations M(k), k ≥ 0 where under each operating mode m ∈ I, the

controller hm(x) is implemented. If Assumption 10 and Assumption 11 hold, and if

x(0) ∈ DM(0), then the set of controllers hm(x), m ∈ I is feasible for all k ≥ 0.

In this chapter, we design EMPC for the scheduled switching operations based on

the set of auxiliary controllers hm(x), m ∈ I, that provides enlarged feasibility region

and practically improved performance over the set of auxiliary controllers.

Remark 6 Note that Assumption 10 and Assumption 11 essentially implies the sta-

bilizability of the optimal steady states of each operating mode of the switched system

of Eq. (4.1), and that the stabilizability of the optimal steady states is not lost between

mode transitions.

4.2 Proposed empc for scheduled switching oper-

ations

In this section, we present the proposed EMPC design for scheduled switching oper-

ations and discuss the recursive feasibility and performance of the proposed EMPC

63

design. In Chapter 2, a terminal cost for the EMPC design for single-mode infinite-

time operation was proposed. The terminal cost is designed such that it characterizes

the economic performance of an auxiliary controller h(x) for Nh steps. Adopting such

a terminal cost extends the prediction horizon of the EMPC design by Nh steps be-

yond the control horizon N . In this chapter, we extend the previous results in [45] to

account for scheduled switching operations. In the proposed EMPC design, we adopt

the same control horizon N while a different terminal cost horizon Nhm is assigned for

each operating mode m ∈ I based on the auxiliary controller hm(x). Specifically, the

terminal cost function for the infinite-time EMPC design under mode m, cm(x,Nhm),

is defined as follows:

cm(x,Nhm) =

Nhm−1∑k=0

l(xhm(k, x), hm(xhm(k, x))) (4.5)

where x ∈ Dm. The cost function cm(x,Nhm) above characterizes the economic per-

formance of the closed-loop system under mode m ∈ I under the controller hm(x) for

Nhm steps with the initial state x ∈ Dm.

4.2.1 Implementation strategy

The implementation of the proposed EMPC design is divided into two phases. In

one phase when the next scheduled mode switching time is beyond the prediction

horizon (i.e., control horizon plus the terminal cost horizon), EMPC takes infinite-

time operation under the current mode; when the next scheduled mode switching

time is within the prediction horizon, EMPC is operated under a mode-switching

phase. The implementation strategy of the proposed EMPC between two consecutive

operating modes m ∈ I and l ∈ I such that moutr + 1 = linq is outlined in Algorithm 1

below:

Algorithm 1

1. Infinite-time operation under mode m. At a sampling time k ∈ [minr , m

outr −

N −Nhm ] when the scheduled mode switching time linq is beyond the prediction

horizon k +N +Nhm , EMPC takes the infinite-time operation of Eq. (4.6).

2. Mode transition operation between m and l

64

2.1 At a sampling time k ∈ [moutr − N − Nhm + 1, mout

r − N + 1] when linq is

within the prediction horizon k+N +Nhm but beyond the control horizon

k +N − 1, EMPC takes the mode-switching operation of Eq. (4.7).

2.2 At a sampling time k ∈ [moutr −N + 2, mout

r ] when linq is within the control

horizon k+N−1, EMPC takes the mode switching operation of Eq. (4.8).

3. Infinite-time operation under mode l. At a sampling time k ∈ [linq , loutq −N−Nhl ],

EMPC takes the infinite-time operation of Eq. (4.6) (with m replaced with l).

A schematic of Algorithm 1 is illustrated in Fig. 4.1. It can be seen that when

Figure 4.1: Time flow of Algorithm 1

an operating mode m is switched in, the proposed EMPC first takes infinite-time

operation for moutr −min

r +1−N−Nhm steps and then takes mode transition operations

for N +Nhm steps before the system is switched to the subsequent operating mode l.

Note that Algorithm 1 implies that the length of an operating mode, moutr −min

r +1 is

no shorter than the prediction horizon, N+Nhm , of the infinite-time EMPC operation

under mode m. This assumption is made explicit below.

Assumption 12 The length of each operating mode m ∈ I in the prescribed schedule

M(k) is no shorter than the prediction horizon of the infinite-time EMPC operation

under the operating mode m. That is, moutr −min

r + 1 ≥ N +Nhm , ∀m ∈ I, ∀r ≥ 1.

Remark 7 Assumption 12 assumes that the length of each operating mode under the

prescribed schedule M(k) is long enough such that each step outlined in Algorithm 1

is carried out. Note that it is a common practice in the control of chemical processes

that the length of an scheduled plan is much longer than the transient settling time of

the system. In this case, at the beginning of an operating mode, the scheduled mode

switching is far away and has little effect on the current control decisions. It is then

a natural strategy to assume infinite-time operation at the beginning and to deal with

mode transition when the scheduled switching time approaches.

65

4.2.2 Proposed EMPC design

At a sampling time k ∈ [minr , m

outr −N −Nhm ], EMPC takes infinite-time operation

under mode m and is formulated as the following optimization problem:

minu(i)

k+N−1∑i=k

lm(x(i), u(i)) + cm(x(N), Nhm) (4.6a)

s.t. x(i+ 1) = fm(x(i), u(i)), i = k, ..., k +N − 1 (4.6b)

x(k) = x(k) (4.6c)

x(i) ∈ Xm, i = k, ..., k +N − 1 (4.6d)

u(i) ∈ Um, i = k, ..., k +N − 1 (4.6e)

x(k +N) ∈ Dm (4.6f)

where x(i) denotes the predicted state trajectory, x(k) is the state measurement at

time instant k. In the above optimization problem, Eq. (4.6a) is the objective function

which is the accumulated economic performance over the prediction horizon N plus

the performance when hm(x) takes over for another Nhm steps under the operating

mode m. Eq. (4.6b) is the system model under the operating mode m. Eq. (4.6c)

specifies the initial condition at time instant k. Eqs. (4.6d) and (4.6e) are the state

and input constraints corresponding to the operating mode m. Eq. (4.6f) requires

that the predicted system state is driven into the terminal region Dm at the end of

the control horizon N .

The infinite-time operation under mode m is carried out until the prediction hori-

zon, k + N + Nhm , reaches the mode switching time moutr . At the time instant

k = moutr − N − Nhm + 1, the EMPC operation is switched into a mode switching

phase which is divided into two stages as outlined in step 2.1 and 2.2 of Algorithm 1.

In the first stage when k ∈ [moutr −N −Nhm + 1, mout

r −N + 1], the proposed EMPC

is formulated as follows:

minu(i)

moutr∑i=k

lm(x(i), u(i)) + cl(x(linq ), Nhl) (4.7a)

s.t. x(i+ 1) = fm(x(i), u(i)), i = k, ...,moutr (4.7b)

x(k) = x(k) (4.7c)

x(i) ∈ Xm, i = k, ..., k +N − 1 (4.7d)

66

u(i) ∈ Um, i = k, ..., k +N − 1 (4.7e)

x(k +N) ∈ Dm (4.7f)

u(i) = hm(x(i)), i = k +N, ...,moutr (4.7g)

x(linq ) ∈ Dl (4.7h)

In the above optimization problem, EMPC optimizes the economic performance under

the operating mode m for N steps before the auxiliary controller hm(x) takes over till

the end of the operating mode m (Eq. (4.7g)). The constraint of Eq. (4.7h) is imposed

to ensure recursive feasibility of the EMPC design for mode switching between the

operating modes m and l. The terminal cost cm(x(linq ), Nhl) is incorporated into the

objective function to further extend the prediction horizon from the operating mode

m into its subsequent operating mode l for Nhl steps. Note that in this stage, the

number of free decision variables in the optimization problem of Eq. (4.7) is constantly

N ×nu, and the interval where the auxiliary controller hm(x) is implemented shrinks

from Nh to 0 as the sampling time k increases.

In the second stage when k ∈ [moutr −N+2, mout

r ], the control horizon covers both

the operating modes m and l. The proposed EMPC design is formulated as follows:

minu(i)

moutr∑i=k

lm(x(i), u(i)) +k+N−1∑i=linq

ll(x(i), u(i)) + cl(x(k +N), Nhl) (4.8a)

s.t. x(i+ 1) = fm(x(i), u(i)), i = k, ...,moutr (4.8b)

x(i+ 1) = fl(x(i), u(i)), i = linq , ..., k +N − 1 (4.8c)

x(k) = x(k) (4.8d)

x(i) ∈ Xm, i = k, ..., koutr (4.8e)

x(i) ∈ Xl, i = linq , ..., k +N − 1 (4.8f)

u(i) ∈ Um, i = k, ..., koutr (4.8g)

u(i) ∈ Ul, i = linq , ..., k +N − 1 (4.8h)

x(k +N) ∈ Dl (4.8i)

Remark 8 Note that the EMPC of Eq. (4.6) for infinite-time operation enjoys an

enlarged operating region than the design in [45] in which the operating region is

restricted to be the stability region of the auxiliary controller. Note also that in the

67

EMPC designs of Eqs. (4.6)-(4.8), the number of free decision variables are all N×nuwhich means that the proposed EMPC scheme for switching operations can be very

computationally efficient compared to that of [3].

Remark 9 It is shown in [45] that the proposed EMPC design for infinite-time op-

eration achieves practically improved asymptotic average performance than the opti-

mal steady state operation. Similar analysis can be applied to the proposed EMPC

of Eq. (4.6). That is, if the terminal cost horizon Nhm is long enough and the cost

lm(x, hm(x)) is sufficiently smooth such that lm(x, hm(x)) ≈ lm(xssm, ussm) for all x ∈ D∗m

where D∗m = xhm(Nhm , x) : x ∈ Dm, then the EMPC of Eq. (4.6) achieves practi-

cally improved average performance than the optimal steady state operation in Step

2.1 of Algorithm 1 given that this phase is long enough.

4.2.3 Recursive feasibility

In this subsection, we analyze the recursive feasibility of the proposed EMPC designs.

Before presenting the results, we need the following assumptions on the length of each

terminal cost horizon Nhm of each operating mode m ∈ I.

Assumption 13 The terminal cost horizon for each operating mode m ∈ I is chosen

such that Nhm ≥ maxN∗ml where N∗ml is defined in Eq. (4.4) and l ∈ I is any

subsequent operating mode of m according to the schedule M(k).

In the following, we show that under the proposed EMPC scheme for scheduled

switching operations, initial feasibility implies recursive feasibility.

Theorem 10 Consider the system of Eq. (4.1) under the prescribed switching sched-

ule M(k) with k ≥ 0. Assume that at a time instant k0 ≥ 0, the proposed EMPC

implementation strategy outlined in Algorithm 1 is switched on and then implemented

for all k ≥ k0. If Assumption 12 and Assumption 13 hold, and if the EMPC design

at k0 is feasible, then the proposed EMPC scheme for scheduled switching operations

remains feasible for all k ≥ k0.

Proof. We provide a sketch of the proof by showing that for any time instant

k ∈ [minr ,m

outr ], the feasibility of the EMPC design at k implies the feasibility of the

EMPC design at k + 1.

68

(1) If the EMPC of Eq. (4.6) is feasible at k = minr , it is recursively feasible for

k ∈ [minr , m

outr − N − Nhm ]. The recursive feasibility of the EMPC of Eq. (4.6) is

a direct consequence of the forward-invariance property of the auxiliary controller

hm(x) inside the terminal region Dm.

(2) If the EMPC of Eq. (4.6) is feasible k = moutr − N − Nhm , then the EMPC

of Eq. (4.7) at k = moutr − N − Nhm + 1 is also feasible. A feasible solution can be

constructed based on the solution at k = moutr −N−Nhm in a standard way. Namely,

by discarding the first element of the optimal input sequence at k = moutr −N −Nhm

and augmenting the sequence by implementing hm(x) for another Nhm+1 steps at the

end of the sequence. Specifically, it can be verified that the constraint of Eq. (4.7f) is

satisfied because of the forward-invariant property of the auxiliary controller hm(x)

inside Dm. Also, the constraint of Eq. (4.7h) is satisfied because of Assumption 12

and that hm(x) is implemented for Nhm + 1 steps in the constructed input sequence.

(3) If the EMPC of Eq. (4.7) is feasible at k = moutr −N−Nhm +1, it is recursively

feasible for k ∈ [moutr −N −Nhm + 1, mout

r −N + 1]. In step 2.1, the EMPC design of

Eq. (4.7) is implemented in a shrinking horizon fashion (the interval where hm(x) is

implemented shrinks ). The solution to a current optimization problem will provide

a feasible solution to the optimization at the next sampling instant by discarding its

first element.

(4) If the EMPC of Eq. (4.7) is feasible at k = moutr − N + 1, then the EMPC

of Eq. (4.8) at k = [moutr − N + 2,mout

r ] is recursively feasible. Again, the standard

approach to construct a feasible solution for the optimization problem at k + 1 can

be applied which is to discard the first element of the optimal input sequence at k

and augment one more input at the end by implementing hl(x) one more step.

(5) If the EMPC of Eq. (4.8) at k = moutr is feasible, then the EMPC of Eq. (4.6)

at k = linq (with m replace with l) is also feasible. Again, the same approach to

construct a feasible solution based on hl(x) can be applied.

Thus, for any time instant k ∈ [minr ,m

outr ], the feasibility of the EMPC design at

k implies the feasibility of the EMPC design at k + 1. Since the mode m ∈ I and

r ≥ 1 are arbitrarily chosen, the above analysis can be applied to every time instant

k ≥ 0 under the prescribed schedule M(k). Therefore, under the proposed EMPC

scheme, initial feasibility implies recursive feasibility.

69

4.3 Application to a chemical process

Consider a well-mixed, non-isothermal continuous stirred tank reactor (CSTR) in

which two irreversible, second-order, endothermic reactions A → B and B → C

take place. Due to the nonisothermal nature of the reactor, a jacket is used to

provide heat to the reactor. The dynamic equations describing the behavior of the

reactor, obtained through material and energy balances under standard modeling

assumptions, are given below:

dCAdt

=F

V(CA0 − CA)− k1e

−E1RT C2

A

dCBdt

= −FVCB + k1e

−E1RT C2

A − k2e−E2RT C2

B

dCCdt

= −FVCC + k2e

−E2RT C2

B

dT

dt= −F

V(T0 − T ) +

−∆H1

ρCpk1e

−E1RT C2

A

+−∆H2

ρCpk2e

−E2RT C2

B +Q

ρCpV

(4.9)

where CA, CB and CC denote the concentrations of the reactant A, product B and

product C; T denotes the temperature of the reactor; Q denotes the heat supply to

the reactor; V represents the volume of the reactor, ∆H1, k1, E1 and ∆H2, k2, E2

denote the enthalpy, pre-exponential constant and activation energy of the reactions

A→ B and B → C, respectively; Cp and ρ denote the heat capacity and the density

of the fluid in the reactor, respectively. The values of the process parameters used

in the simulations are shown in Table 6.2. The process model of Eq. (4.9) belongs

to the following class of nonlinear systems: x = f(x, u) where the state vector is

x = [CA, CB, CC , T ]T , the control input is u = Q.

In this example, two operating modes are specified. In mode 1, the objective

function is l1 = −CB, which is to maximize the product of B. In mode 2, the

objective function is l2 = −CC , which is to maximize the product of C. Under the

two operating modes, the system models and the input and state constraints remain

the same. Specifically, f1 = f2 = f , X1 = X2 = R4, U1 = U2 = u : [0, 5×105] kJ/h.

By solving the steady-state optimization problem of Eq. (4.2), we obtain the opti-

mal steady states under the two operating modes as follows: xss1 = [1.35, 1.98, 0.67, 188.92],

70

Table 4.1: Model parameters.

T0 = 300 K k1 = 176.94 1h

CA0 = 4 kmolm3 k2 = 10.88 1

h

V = 1 m3 E1 = 5000 kJkmol

F = 5 m3

hE2 = 4000 kJ

kmol

Cp = 0.231 kJkgK

∆H1 = 1.15× 104 kJkmol

ρ = 1000 kgm3 ∆H2 = 1.05× 104 kJ

kmol

R = 8.314 kJkmolK

uss1 = 5.95×104 and xss2 = [0.61, 1.59, 1.80, 448.63], uss2 = 5×105. Since the system of

Eq. (4.9) is open-loop stable, we use the open-loop optimal steady-state inputs as the

auxiliary controllers with h1(x) = uss1 and h2(x) = uss2 . The two auxiliary controllers

are globally stabilizing with D1 = D2 = R4.

The system of Eq. (4.9) is discretized with a sampling time ∆ = 0.1h. The

operating policy for the time interval of t ∈ [0, 6h], or k ∈ [0, 59], is scheduled as

follows:

M(k) = 1, 0 ≤ k ≤ 29

2, 30 ≤ k ≤ 59

In the simulations, we compare the proposed EMPC scheme for scheduled switching

operations with the approach in [3] as well as the auxiliary controllers. The ini-

tial state is x(0) = [0.61, 1.59, 1.80, 448.63]. The closed-loop state and input trajec-

tories under the three approaches are shown in Fig. 4.2 and Fig. 4.3 respectively.

Let us define the following performance indexes to better compare the perfor-

mance of each control configurations. The performance under operating mode 1

for t ∈ [0h, 3h) is J1 =∑29

k=0 CB(k). The performance under operating mode 2

for t ∈ [3h, 6h] is J2 =∑59

k=30CC(k). The overall performance for t ∈ [0h, 6h] is

Jall = J1 + J2. The performance during the mode switching for t ∈ [2.5h, 3.5h] is

Jswitch =∑29

k=25CB(k) +∑34

k=30CC(k). These performance indexes under the three

control configurations are shown in Table 4.2. In addition, the average controller

evaluation times of the proposed approach and that of the approach in [3] are shown

in Table 4.3

From these results, it can be seen that the three control configurations provide

71

0 1 2 3 4 5 60.5

1

1.5

CA

(km

ol/m

3)

0 1 2 3 4 5 61.5

2

2.5

CB

(km

ol/m

3)

0 1 2 3 4 5 60

1

2

CC

(km

ol/m

3)

0 1 2 3 4 5 60

500

t (h)

T (

K)

Figure 4.2: Closed-loop state trajectories of the proposed approach (solid lines), theapproach in [3] (dashed lines) and the auxiliary controllers (dash-dotted line).

approximately the same performance for single-mode operation (i.e., when a current

operating time is far away from the scheduled mode switching time) while the main

differences occur during the mode transition operations. The EMPC proposed in [3]

achieves the optimal performance under operating mode 1 for t ∈ [0h, 3h) because

it is initialized with a control horizon of the length of the operating mode 1 and

is implemented in a shrinking horizon fashion. However, the performance between

mode switching is compromised since it is not directly accounted for in the EMPC

in [3]. The proposed EMPC design for scheduled mode switching operations provide

the best performance during the mode switching operations and the best overall

performance. This is mainly due to the extended prediction horizon based on the

auxiliary controllers to account for the performance between mode transitions. It

can be also seen from Table 4.3 that the proposed approach is more computationally

efficient compared to the EMPC design in [3].

72

0 1 2 3 4 5 60

1

2

3

4

5x 10

5

t (h)

Q (

kJ/h

)

Figure 4.3: Closed-loop input trajectories of the proposed approach (solid line), the ap-proach in [3] (dashed line) and the auxiliary controllers (dash-dotted line).

Table 4.2: Performance under different configurations

EMPC in [3] Auxiliary controllers Proposed

J1 57.6 57.8 57.7

J2 55.9 54.3 53.2

Jall 113.5 112.1 110.9

Jswitch 18.8 17.6 16.4

Table 4.3: Average controller evaluation time

Proposed EMPC in [3]

time(s) 0.8 18.1

73

Chapter 5

Nonlinear MPC for zone tracking

5.1 Introduction

Model predictive control (MPC) is the most widely applied advanced control tech-

nique in the process industry due to its many advantages such as optimally handling

process constraints and interaction. In the traditional paradigm, MPC fulfills the

objective of set-point tracking by penalizing the deviation of the predicted state and

input trajectories to the desired set-point. In practice, it is often beneficial to de-

sign MPC that tracks a zone region instead of a set-point. The superiority of zone

control over set-point control are two-folds: first, it is capable of flexibly handling

multiple objectives so that the degrees of freedom in the controller can be assigned to

address more important objectives; second, it is inherently robust in the presence of

model mismatch or process disturbances. As a matter of fact, all industrial MPCs are

equipped with zone control option to some extent [4]. Successful applications of zone

MPC have been reported in various areas such as diabetes treatment [46], control of

building heating system [47], and control of the fluid catalytic cracking unit in oil

refinery [48].

Despite the widely successful industrial applications, there has not been a sys-

tematic approach for the design and analysis of MPC for zone tracking. On the one

hand, reported industrial zone MPC designs are heuristic in nature and lack stability

guarantee. On the other hand, existing zone MPC designs with guaranteed stability

[49], [50] essentially convert the zone tracking objective to tracking of the steady-

state subspace in the target zone. The trick is to cast the steady-state set-point also

as decision variables in the set-point tracking MPC scheme. However, tracking of

74

a target zone is not equivalent to tracking of its steady-state subspace. As far as

the satisfaction of a zone target is concerned, the system does not necessarily take

steady-state operation.

Several difficulties arise when it comes to the design and analysis of MPC which

tracks a zone target in a straightforward manner. First of all, the admissibility of

a zone target needs to be carefully examined. Since the target zone is often user-

specified based on the economic metrics of the process, it is not necessarily positive

invariant. One might be tempted to find the largest positive invariant set in the target

zone and convert the zone tracking objective to tracking of its largest positive invariant

subset. Unfortunately, finding and characterizing the maximal positive invariant set

in the target zone can be very difficult or expensive even for linear systems [51]. Even

if the largest positive can be found, the aforementioned approach may not be desirable

and may give deteriorated transient performance as shown by our simulation study.

In this work, we propose a general framework for nonlinear model predictive zone

control. The proposed zone MPC tracks a target zone characterized by coupled sys-

tem state and input. A control invariant subset of the target zone is incorporated

in the proposed zone MPC as the terminal constraint to ensure closed-loop stability.

We resort to the invariance principle and develop invariance-like theorem which is

suitable for stability analysis of zone control. It is proved that under the proposed

zone MPC design, the system converges to the largest control invariant subset of the

target zone. In the stability analysis, we focus on the evolvement of the state-input

pair (x(n), u(n)) instead of merely the state trajectory x(n). This provides a better

picture and more accurate description of the system behavior, especially when the

input trajectory is also of interest. Further discussions are made on enlargement of

the region of attraction by employing an auxiliary controller as well as handling a

secondary economic objective via a second-step economic optimization. Two numer-

ical examples are used to demonstrate the superiority of zone control over set-point

control and the efficacy of the zone MPC design.

75

5.2 Problem setup

5.2.1 Notation

Throughout this work, the operator | · | denotes the Euclidean norm of a scalar or a

vector. The symbol |x|A := infz∈A|x − z| denotes the distance of a point x to the set

A. The symbol Bε(X) := x : |x|X < ε denotes the set of points whose distance to

the set X is less than ε. The symbol ‘\’ denotes set substraction such that A \ B :=

x ∈ A, x /∈ B. Vε(x) denotes the ε sublevel set of the function V (x) such that

Vε(x) = x : V (x) ≤ ε. The symbol INM denotes the set of integers M,M+1, ..., N.

The symbol I≥0 denotes the set of non-negative integers 0, 1, 2, .... The symbol

projX(O) denotes the projection of the set O onto its subspace X. A continuous

function α : [0, a) → [0,∞) is said to belong to class K if it is strictly increasing

and satisfies α(0) = 0. A class K function α is called a class K∞ function if α is

unbounded.

5.2.2 System description and control objective

We consider the following nonlinear discrete time system:

x(n+ 1) = f(x(n), u(n)) (5.1)

where x(n) ∈ X ⊂ Rnx , u(n) ∈ U ⊂ Rnu , n ∈ I≥0, denote the state and input at

time n, respectively. The vector function f(·) : Rnx × Rnu → Rnx is continuous. The

system is subject to coupled state and input constraint: (x, u) ∈ Z ⊆ X × U, where

X, U, Z are all compact sets. The control objective is to steer and maintain the

system in a compact set Zt ⊂ Z. The target set Zt may correspond to a desired range

of process state, input or output specified by user based on economic metrics of the

process. We say that the target set Zt is admissible if there exist complete state and

input trajectories bounded in Zt such that (x(n), u(n)) ∈ Zt, n ∈ I≥0. To address the

admissibility of the zone control objective, we need the following definition:

Definition 3 (control invariant set) A set O ⊆ X× U is called control invariant or

a control invariant set if any (x, u) ∈ O implies f(x, u) ∈ projX(O).

Since the target set Zt is user-defined based on economic metrics of the process, it is

not necessarily control invariant. A reasonable assumption we make in this work is

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that the target set Zt is admissible. This is equivalent to assuming the existence of

a nonempty control invariant subset of Zt. Apparently, any admissible steady-state

pair (xs, us) ∈ Zt with xs = f(xs, us) constitutes a control invariant subset of Zt. The

conventional set-point control can be viewed as a special case of zone control where

the target set degenerates to a singleton Zt = (xs, us).

Remark 10 In the classical definition, the control invariant set refers to a set of

system state which is positive invariant under certain control policy (e.g., [32] Defi-

nition 2.10). In Definition 3, the control invariant set O ⊆ X × U is defined in the

augmented state-input space. We can think of the control invariant set O as induced

by the control invariant state set projX(O) in the classical definition which is positive

invariant under the set-valued control policy (x, u) ∈ O. The reason to consider the

augmented state-input space will become clear along the discussion as it provides a

better picture and more accurate description of system behavior for zone control.

5.3 Zone MPC formulation

At a sampling time n, the proposed zone MPC is formulated as the following opti-

mization problem:

V 0N(x(n)) = min

ui,xzi ,uzi

N−1∑i=0

|xi − xzi |2Q + |ui − uzi |2R (5.2a)

s.t. xi+1 = f(xi, ui), i ∈ IN−10 (5.2b)

x0 = x(n) (5.2c)

(xi, ui) ∈ Z, i ∈ IN−10 (5.2d)

(xzi , uzi ) ∈ Zt, i ∈ IN−1

0 (5.2e)

xN ∈ Xf (5.2f)

The objective function Eq. (5.2a) penalizes the distance from the predicted state and

input trajectories to the target zone Zt by introducing artificial variables xzi and uzi

bounded by Eq. (5.2e). Q, R are positive-definite weighting matrices. The terminal

set Xf ⊆ projX(Zt) is designed to ensure recursive feasibility and stability of the zone

MPC design. Specifically, Xf satisfies the following control invariant condition:

77

Assumption 14 For any x ∈ Xf , there exists a u such that (x, u) ∈ Zt and f(x, u) ∈

Xf

Let u∗(i+n|n), i ∈ IN−10 , denote the optimal solution to Eq. (5.2). The corresponding

state trajectory is x∗(i + n|n), i ∈ IN−10 . The input injected to the system at time

n is: u(n) = u∗(n|n). At time n + 1, the optimization of Eq. (5.2) is re-evaluated.

Note that the solution of u∗(n|n) may not be unique because of the nature of the

zone tracking objective. This means that both the input and state trajectories of the

closed-loop system may be set-valued.

5.4 Stability analysis

In this section, we discuss the stability of the proposed zone MPC design. Several

difficulties arise when it comes to the stability analysis of zone MPC. First, the target

set Zt ⊂ X×U involves zone regions for both system state and input. Unlike set-point

control where asymptotic tracking of the steady state (i.e., x(n)→ xs) usually implies

asymptotic tracking of the corresponding steady-state input (i.e., u(n)→ us), in zone

control it is either insufficient or inappropriate to only consider the evolvement of the

state trajectory x(n). The trajectory (x(n), u(n)) in the augmented state-input space

should be investigated. Second, the target set Zt may not be control invariant. This

means that we cannot establish Lyapunov stability for the target set Zt by finding

a suitable Lyapunov function. Instead, we will resort to the more general tools by

LaSalle’s invariance principle.

In this section, we will first develop invariance-like theorem which generalizes

LaSalle’s invariance principle to cope with the analysis of control systems of Eq. (5.1)

based on the control invariant set as defined in Definition 3. Then we establish

stability of the proposed zone MPC design. It is proved that the system converges to

the largest control invariant set in the target set Zt.

Definition 4 (Accumulation point and accumulation set) A point x is called an ac-

cumulation point of the sequence x(n) if there is an increasing sequence of integers

ni, i ∈ I≥0, such that limi→∞

x(ni) = x. The accumulation set Sx(n) of a sequence x(n)

is the set of all accumulation points of x(n). ∗

∗In the context of autonomous dynamical systems (i.e., xn+1 = f(xn)), accumulation point and

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Definition 5 (Attractive set) A set S is called attractive or an attractive set of a

sequence x(n) if: limn→∞

|x(n)|S = 0.

In short, the accumulation point of a sequence x(n) is the limit of a subsequence of

x(n). The attractive set of x(n) is the set that x(n) approaches as n → ∞. For

notational simplicity, we will use x(n) → S to denote that S is attractive. The

following result is a simplification of Theorem 5.2 in [52]. We provide the proof here

for the completeness of this work.

Lemma 17 If the sequence x(n) is bounded in a compact set X for all n ∈ I≥0, then

the accumulation set Sx(n) is nonempty, compact, and is the smallest closed attractive

set of x(n).

Proof. Since x(n) is bounded in X, by the Bolzano – Weierstrass Theorem, x(n)

contains a convergent subsequence. Hence Sx(n) is nonempty. It is also easy to show

that Sx(n) is closed by showing that its complement X \ Sx(n) is open. Since Sx(n) is

bounded in X and closed, it is compact. Next Assume that Sx(n) is not attractive,

then there exists x /∈ Sx(n) such that x is an accumulation point of x(n). This is a

contradiction because of the definition of Sx(n). Thus, Sx(n) is attractive. Assume

that there is an attractive set S of x(n) which is a strict subset of Sx(n), then there

exists an accumulation point x ∈ Sx(n) such that |x|S > 0. This implies that S is not

attractive, also a contradiction. Therefore Sx(n) is the smallest closed attractive set

of x(n).

Lemma 17 holds for any bounded sequence which is not necessarily a state or input

trajectory. For the state and input trajectories of the control system of Eq. (5.1), we

have the following result:

Theorem 11 Consider the control system of Eq. (5.1), if the corresponding state

and input trajectories are bounded such that (x(n), u(n)) ∈ Z for all n ∈ I≥0, then the

accumulation set S(x(n),u(n)) is control invariant.

accumulation set are often referred to as limit point and limit set. However, since we are consideringcontrol systems (i.e. xn+1 = f(xn, un)) where the control law un = κ(xn) might be set-valued ordiscontinuous, we choose the terms accumulation point and accumulation set to avoid confusion withexisting results such as LaSalle’s invariance principle whose proof relies on the continuity of f andthe invariant limit set.

79

Proof. From Lemma 17 we know that the accumulation set S(x(n),u(n)) is non-empty

and compact. We prove Theorem 11 by contradiction. Assume that S(x(n),u(n)) is not

control invariant, then there is an accumulation point (x∗, u∗) ∈ S(x(n),u(n)) such that

|f(x∗, u∗)|Sx(n) > 0 where Sx(n) is the limit set of x(n) (i.e., the projection of S(x(n),u(n))

onto X). Let ε = |f(x∗, u∗)|Sx(n) > 0. Because of the continuity of f(·), there exists

δ > 0 such that:

|f(x, u)|Sx(n) > ε/2, ∀(x, u) ∈ Bδ((x∗, u∗)) (5.3)

Because (x∗, u∗) is an accumulation point, there is an increasing sub-sequence

(x(ni), u(ni)) such that (x(ni), u(ni)) ∈ Bδ((x∗, u∗)) for all i ∈ I≥0. However,

from Eq. (5.3), this implies that |x(ni + 1)|Sx(n) > ε/2 for all i ∈ I≥0, which furthers

implies the existence of an accumulation point outside Sx(n). This contradicts the

definition of the accumulation set Sx(n). Therefore, the accumulation set S(x(n),u(n)) is

control invariant.

Let ZMt denote the maximal control invariant subset of the target set Zt ⊂ Z.

That is,

ZMt := (x, u) | (x, u) ∈ Zt, f(x, u) ∈ projX(Zt)

By definition, ZMt is the union of all control invariant subsets of Zt. The follow-

ing theorem generalizes LaSalle’s invariance principle from autonomous systems (i.e.,

xk+1 = f(xk)) to control systems (i.e., xk+1 = f(xk, uk)) based on the definition of

control invariant set in the augmented state-input space in Definition 3.

Theorem 12 Consider the control system of Eq. (5.1) with (x(n), u(n)) ∈ Z for all

n ∈ I≥0.

(i) If there is a bounded function V (x) : X→ R and a continuous function ρ(x, u) :

Z→ R≥0 which is positive-definite with respect to Zt such that the following holds:

V (x(n+ 1))− V (x(n)) ≤ −ρ(x(n), u(n)) (5.4)

for all n ∈ I≥0, then (x(n), u(n))→ ZMt .

(ii) If, in addition, V (x) is locally continuous on projX(Zt) and constant on

projX(ZMt ), then ZMt is asymptotically stable.

80

Proof. (i) First we prove by contradiction that S(x(n),u(n)) ⊆ Zt. Assume that there

is an accumulation point (x∗, u∗) /∈ Zt such that ρ(x∗, u∗) = ε > 0. Using sim-

ilar arguments as in the proof of Theorem 11, there is an increasing subsequence

(x(ni), u(ni)) → (x∗, u∗), i ∈ I≥0, such that ρ(x(ni), u(ni)) > ε/2. However, from

Eq. (5.4), this implies that V (·) is decreased by ε/2 infinitely many times which con-

tradicts the boundedness of V (·). Therefore, the sequence (x(n), u(n)) cannot have

an accumulation point outside Zt, which equivalently means that S(x(n),u(n)) ⊆ Zt.

Applying Theorem 11 and based on the definition of ZMt , we have S(x(n),u(n)) ⊆ ZMt .

From Lemma 17, (x(n), u(n))→ S(x(n),u(n)), thus (x(n), u(n))→ ZMt .

(ii) Summing Eq. (5.4) for n ∈ IN−10 , we have:

V (x(0)) ≥ V (x(N)) +N−1∑n=0

ρ(x(n), u(n)) (5.5)

Let V (x) = c for x ∈ projX(ZMt ). Since (x(n), u(n)) → ZMt and V (x) is locally

continuous on projX(Zt), Eq. (5.5) when taking limit as N →∞ becomes:

V (x(0)) ≥ c+∞∑n=0

ρ(x(n), u(n)) (5.6)

Let us define a function V ′(x, u) : Z → R≥0 such that V ′(x, u) = V (x) − c. V ′(x, u)

is locally continuous on Zt with V ′(x, u) = 0 for (x, u) ∈ ZMt . Substituting V (x) by

V ′(x, u), Eq. (5.6) becomes:

V ′(x(0), u(0)) ≥∞∑n=0

ρ(x(n), u(n))

It is easy to see from the above that V ′(x, u) > 0 for (x, u) /∈ ZMt because any

(x(0), u(0)) /∈ ZMt implies the existence of (x(M), u(M)) /∈ Zt for some M so that

V ′(x(0), u(0)) ≥ ρ(x(M), u(M)) > 0. Thus V ′(x, u) is positive-definite with respect to

ZMt . Moreover, substituting V ′(x, u) into Eq. (5.4), we have that V ′(x(n + 1), u(n +

1)) − V ′(x(n), u(n)) ≤ −ρ(x(n), u(n)) where ρ(x(n), u(n)) is positive semidefinite

with respect to ZMt . These properties make V ′(x, u) a Lyapunov function for ZMt .

Therefore ZMt is stable in the sense of Lyapunov. Since it is also attractive from (i),

ZMt is asymptotically stable.

Before stating and proving the stability results of the proposed zone MPC design,

we need to define the N -step controllable set as follows:

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Definition 6 (N-step controllable set [53]) We use XN(Z,Xf ) to denote the set of

states in X that can be steered to Xf in N steps while satisfying the state and input

constraints (x, u) ∈ Z. That is,

XN(Z,Xf ) =x(0) ∈ projX(Z) | ∃ (x(n), u(n)) ∈ Z, n ∈ IN−1

0 , x(N) ∈ Xf

Theorem 13 Consider the system of Eq. (5.1) under the zone MPC of Eq. (5.2). If

Assumption 14 holds and x(0) ∈ XN(Z,Xf ), then:

(i) The zone MPC is recursively feasible with (x(n), u(n))→ ZMt .

(ii) If, in addition, the value function V 0N(x(n)) is locally continuous on projX(Zt)

and XN(Zt,Xf ) = projX(ZMt ), then ZMt is asymptotically stable.

Proof. (i) From Assumption 14, the terminal set Xf is control invariant under the

constraint (x, u) ∈ Zt and therefore control invariant under the constraint (x, u) ∈

Z. Given the optimal solution at time n − 1, a feasible solution at time n can

be constructed by letting u(i + n|n) = u∗(n + i|n − 1), i ∈ IN−20 and by finding

u(N − 1 + n|n) based on Assumption 14. The result is standard in MPC literature

(see e.g., [5], [32]) and is omitted here for brevity. Also following the standard analysis

for optimal value function decrease, the following can be obtained:

V 0N(x(n+ 1))− V 0

N(x(n)) ≤ −l∗(x(n), u(n)), n ∈ I≥0

where l∗(x(n), u(n)) is the optimal objective function to the following optimization

problem:l∗(x(n), u(n)) = min

xz ,uz|x(n)− xz|2Q + |u(n)− uz|2R

s.t. (xz, uz) ∈ ZtIt is easy to check that l∗(x(n), u(n)) is continuous and positive-definite with respect

to the target set Zt. Thus Eq. (5.4) of Theorem 12 is satisfied with V (x) = V 0N(x)

and ρ(x, u) = l∗(x, u). Apply Theorem 12 (i), we have (x(n), u(n))→ ZMt .

(ii) It is easy to check from the definition of the N -step controllable set that

V 0N(x) = 0 for all x ∈ XN(Zt,Xf ). Since XN(Zt,Xf ) = projX(ZMt ), V 0

N(x) = 0 for all

x ∈ projX(ZM). Apply Theorem 12 (ii), ZMt is asymptotically stable.

Remark 11 In the stability analysis of MPC, (local) continuity of the value function

V 0N(x(n)) is usually required to establish asymptotic stability. Interested readers may

82

refer to [54] or [32] Appendix C.3 for more detailed discussions on sufficient conditions

for continuous value function. The key idea is to examine the continuity of the set-

valued map UN(x) : XN(Zt,Xf )→ RN×nu where

UN(x(n)) :=u0, u1, ..., uN−1 | x(n) ∈ XN(Zt,Xf ),Eq. (5.2) is feasible

If UN(x(n)) is continuous on int (XN(Zt,Xf )), then V 0

N(x(n)) is continuous on

int (XN(Zt,Xf )). A special case where UN(x(n)) is guaranteed to be continuous

is linear system (f) with convex constraints (Z,Xf). In this case, if projX(Zt) ⊂

int(XN(Zt,Xf )), then V 0N(x(n)) is locally continuous on projX(Zt).

5.5 Further discussions

5.5.1 Enlargement of the region of attraction

From Theorem 13, the largest control invariant set ZMt in the target zone Zt is at-

tractive with a region of attraction XN(Z,Xf ). From the definition of the N -step

controllable set XN(Z,Xf ), two straightforward ways to enlarge the region of attrac-

tion are (a) to increase the control horizon N , and (b) to use a larger control invariant

terminal set Xf . The former is always a possible choice with its applicability limited

by the online computation power. The latter amounts to finding the largest control

invariant state set in the target zone (i.e., projX(ZMt )) which is difficult for generic

nonlinear control systems. Interested readers may refer to [55], [56], [57], [58] for some

existing results on the computation and approximation of the maximal invariant set.

Another approach which has proven very effective is to employ an auxiliary control

law in the MPC design [59], [23]. By extending the prediction horizon based on the

auxiliary controller, enlargement of the region of attraction as well as improvement

of transient performance can be achieved without significantly increasing the compu-

tational complexity. In the following we discuss how the methodology can be applied

in the MPC design for zone tracking. We make the following assumption about the

auxiliary control law.

Assumption 15 There is a continuous control law u = h(x) which uniformly asymp-

totically stabilizes a positive invariant set Xt,h ⊂ X with (x, h(x)) ∈ int(Zt), ∀x ∈ Xt,h.

The region of attraction Xh ⊂ X is such that projX(Zt) ⊂ Xh ⊂ projX(Z).

83

Let xh(k, x), uh(k, x) denote the closed-loop state and input trajectories under the

controller h(x) at time instant k with xh(0, x) = x and uh(k, x) = h(xh(k, x)). More-

over, let XMt,h ⊆ X denote the largest positive invariant set under the controller h(x)

in the target zone Zt. That is,

XMt,h := x | (xh(k, x), uh(k, x)) ∈ Zt, ∀k ∈ I≥0

From the definition of XMt,h and ZMt , we have XM

t,h ⊆ projX(ZMt ). The following result

will be employed in the zone MPC design:

Lemma 18 If Assumption 15 holds, then there exists Nh ∈ I≥0 such that xh(k, x) ∈

XMt,h for all k ≥ Nh and x ∈ Xh.

Proof. From the converse Lyapunov thoerem [60], there exists a smooth Lyapunov

function V (xh(k, x)) and K∞ functions α1, α2 and α3 such that the following holds

for all k ≥ 0 and x ∈ Xh:

α1(|xh(k, x)|Xt,h) ≤ V (xh(k, x)) ≤ α2(|xh(k, x)|Xt,h)

V (xh(k + 1, x))− V (xh(k, x)) ≤ −α3(|xh(k, x)|Xt,h)

Since V (·) is continuous and (x, h(x)) ∈ int(Zt) for x ∈ Xt,h, there exists a ε sublevel

set of the Lyapunov function, Vε(x), with ε > 0 such that (x, h(t)) ∈ Zt for all

x ∈ Vε(x). Moreover, Vε(x) is positive invariant under h(x) (because of the non-

increasing property of V (·)) and is therefore a subset of XMt,h. Since Xt,h is a strict

subset of Vε(x) and is uniformly asymptotically stable under h(x), there is Nh ∈ I≥0

such that xh(k, x) ∈ Vε(x) for all k ≥ Nh and x ∈ Xh. Thus, xh(k, x) ∈ Vε(x) ⊆ XMt,h

for all k ≥ Nh and x ∈ Xh.

The results of Lemma 18 is illustrated in Figure 5.1. In the availability of the

auxiliary controller h(x), the zone MPC of Eq. (5.2) can be reinforced as follows:

V 0N(x(n)) = min

ui,xzi ,uzi

N+Nh−1∑i=0

|xi − xzi |2Q + |ui − uzi |2R (5.7a)

s.t. xi+1 = f(xi, ui), i ∈ IN−10 (5.7b)

x0 = x(n) (5.7c)

(xi, ui) ∈ Z, i ∈ IN−10 (5.7d)

84

x

xh(N

h,x)

Xt,h

XMt,h

projX

(Zt)

Xh

X

Figure 5.1: Closed-loop state trajectory under controller u = h(x): for any initial statex ∈ Xh, the trajectory xh(k, x) enters the positive invariant set XMt,h in Nh steps and ismaintained in it thereafter.

(xzi , uzi ) ∈ Zt, i ∈ IN+Nh−1

0 (5.7e)

xN ∈ Xh (5.7f)

ui = h(xi), i ∈ IN+Nh−1N (5.7g)

The above zone MPC design utilizes the auxiliary controller h(x) to extend the predic-

tion horizon of the zone MPC of Eq. (5.2) for another Nh steps. If the terminal horizon

Nh is chosen such that it satisfies the condition of Lemma 18, then the objective func-

tion of Eq. (5.7a) is equivalent to an infinite horizon cost (with (xi, ui) = (xzi , uzi ) ∈ Zt

for i ≥ N +Nh). Stability of the zone MPC of Eq. (5.7) is summarized as follows:

Theorem 14 Consider the system of Eq. (5.1) under the zone MPC of Eq. (5.7) with

x(0) ∈ XN(Z,Xh). If Assumption 15 holds and the terminal horizon Nh is chosen

such that it satisfies the condition of Lemma 18, then:

(i) The zone MPC is recursively feasible with (x(n), u(n))→ ZMt .

(ii) If, in addition, the value function V 0N(x(n)) is locally continuous on projX(Zt)

and

XN(Zt,XMt,h) = projX(ZMt ), then ZMt is asymptotically stable.

Proof. The proof of Theorem 14 (i) is similar to the proof of Theorem 13 (i) with

the feasible solution for u(N − 1 + n|n) constructed by u(N − 1 + n|n) = h(x∗(N −

85

1 + n|n − 1)) based on Assumption 15. The proof of Theorem 14 (ii) is similar to

Theorem 13 (ii) with XN(Zt,Xf ) replaced by XN(Zt,XMt,h). The details are omitted

for brevity.

Remark 12 Comparing Theorem 13 (i) and Theorem 14 (i), we can see that the

region of attraction is enlarged from XN(Z,Xf ) to XN(Z,Xh) (because Xf ⊂ Xh).

Moreover, from Theorem 13 (ii) and Theorem 14 (ii), we can see that larger terminal

set Xf or XMt,h is desirable as far as the size of the stability region in the target set is

concerned. We could theoretically use XMt,h as the terminal constraint set Xf in the

zone MPC design of Eq. (5.2). Unfortunately, given the auxiliary controller h(x) and

the target set Zt, the largest positive invariant set XMt,h can be very difficult or compu-

tationally expensive to characterize. Employing the auxiliary controller explicitly in

the zone MPC design as in Eq. (5.7) spares us the burden of explicitly characterizing

XMt,h.

5.5.2 Handling a secondary economic objective

If the optimal solution to the optimization problem of Eq. (5.2) is non-unique, an

economic optimization can be carried out subsequently to make use of the rest degrees

of freedom. Specifically, the optimization problem is formulated as follows:

minui,xzi ,u

zi

N−1∑i=0

l(xi, ui) (5.8a)

s.t. (5.2b)− (5.2f) (5.8b)

N−1∑i=0

|xi − xzi |2Q + |ui − uzi |2R = V 0N(x(n)) (5.8c)

where the cost function l(xi, ui) : Z→ R characterizes process economic performance.

Constraints (5.8b) and (5.8c) confine the solution space of the economic optimization

to be in the solution space of the zone MPC of Eq. (5.2). Therefore the stability

result of Theorem 13 is inherited.

It is also interesting to view the optimization problem of Eq. (5.8) as an economic

MPC (EMPC). In particular, if V 0N(x(M)) = 0 at some time instant M ∈ I≥0, then

86

for all n ≥M , Eq. (5.8) is equivalent to the following EMPC design:

minui

N−1∑i=0

l(xi, ui)

s.t. xi+1 = f(xi, ui), i ∈ IN−10

x0 = x(n)(xi, ui) ∈ Zt, i ∈ IN−1

0

xN ∈ Xf

(5.9)

In the above EMPC optimization problem, the predicted state and input trajectories

are constrained by the target set Zt. Existing EMPC design and analysis methods

(see e.g., [61], [10]) can be applied to the EMPC of Eq. (5.9). For example, one can

find the optimal steady state with respect to the economic cost function l(x, u) in

the target set Zt and verify the dissipativity condition of the optimal steady state

in Zt. Likewise, if the auxiliary controller is utilized in the zone MPC design as

in Eq. (5.7), the corresponding secondary economic optimization problem can be

designed similarly. Existing results of EMPC based on an auxiliary controller [59],

[45] can be applied.

5.6 Examples

Example 1. Consider the following linear scalar system:

x(n+ 1) = 1.25x(n) + u(n)

with state and input constraints X = [−5, 15], U = [−5, 5] respectively. The target

set is Zt = (x, u) | x ∈ [−5, 15], u ∈ [−1, 1]. Using the algorithms in [53] [62],

the largest control invariant set in the target zone can be obtained as follows: ZMt =

(x, u) | Gx+Hu ≤ 1 where G = [0, 0, 0.3125,−0.3125]T , H = [1,−1, 0.25,−0.25]T

(Figure 5.2).

In the simulation, we compare four different zone MPC configurations: (a) a zone

MPC that penalizes the distance to the target set Zt (without terminal constraint),

(b) a zone MPC that penalizes the distance to the largest control invariant set in

the target zone ZMt (also without terminal constraint), (c) the proposed zone MPC

of Eq. (5.2) with Xf = projX(ZMt ) = [−4, 4], and (d) the proposed zone MPC of

Eq. (5.7) with the auxiliary control law h(x) = −0.3x (which asymptotically stabilizes

87

the origin), Xh = [−5, 12], Nh = 20. It is easy to check that the condition of Lemma 18

is satisfied.

The control horizon of the zone MPCs (a), (b), (c) is N = 10 whereas for zone

MPC (d), a short control horizon N = 3 is used. The simulation results are shown in

Figure 5.2. The overall transient performance of the four different zone MPC designs

over 30 sampling times J0:29 =∑29

k=0 |(xk, uk)|Zt are shown in Table 5.1.

-5 0 5 10 15

x

-5

0

5

u

ZtM Z

tZ

Figure 5.2: State and input trajectories of a): a zone MPC that tracks the target zone Zt(diamonds), b): a zone MPC that tracks the largest control invariant set in the target zoneZMt (squares), c): the proposed zone MPC of Eq. (5.2) (triangles), and d): the proposedzone MPC of Eq. (5.7) (circles).

Table 5.1: The transient performance of the four zone MPC configurations over 30 sam-pling times

Zone MPC (a) Zone MPC (b) zone MPC (c) zone MPC (d)J0:29 70.19 63.55 56.65 56.72

From these results, we can see that the zone MPC (a) which penalizes the dis-

tance to the target set Zt without terminal conditions fails to asymptotically track

the target set Zt. The zone MPC (b) that penalizes the distance to the largest control

invariant set in the target zone ZMt achieves its tracking objective, but with deterio-

rated transient performance as compared to the proposed zone MPC design (c) and

88

(d). Moreover, a closer look at the region of attractions of the proposed zone MPC de-

signs (c) and (d) reveals that when N = 3, x(0) /∈ XN(Z,Xf ) and x(0) ∈ XN(Z,Xh).

This indicates that by employing the auxiliary control law in the zone MPC design,

we are allowed to use smaller control horizon N .

It is also worth pointing out that the above results resemble the comparison be-

tween set-point MPC and several popular economic MPC (EMPC) designs [28], [10],

[59]. Specifically, zone MPC (a), (b) corresponds to EMPC without terminal con-

straint and set-point MPC, respectively [10]. Zone MPC (c) corresponds to EMPC

with a point-wise terminal constraint [28]. Zone MPC (d) is analogous to EMPC

with extended horizon based on an auxiliary controller [59].

Example 2. Consider the following nonlinear scalar system:

x(n+ 1) = x(n)2 + u(n) (5.10)

with initial condition x(0) = −1 and input constraint U = [−1.05,−0.95]. The

primary control objective is to maintain the system operation in the target zone:

Zt = (x, u) : x ∈ [−1.5, 0.5], u ∈ [−1.05,−0.95]. There is also a secondary economic

objective to minimize the control input l(x, u) = u. The corresponding optimal steady

state is (xs, us) = (−0.6402,−1.05). We compare two control configurations: (a) a set-

point MPC which tracks (xs, us) (without terminal constraint), and (b) the proposed

zone MPC of Eq. (5.2) which handles the economic objective by taking the second

step economic optimization of Eq. (5.8). The terminal constraint for the zone MPC is

Xf = [−1.5, 0.5]. Both MPCs take the same control horizon N = 5. The simulation

results of cases (a) and (b) are shown in Figure 5.3 and Figure 5.4, respectively.

From Figure 5.3 and Figure 5.4, we can see that set-point MPC achieves its

tracking objective at the expense of making excessive control moves. Note that this

is not because of poor parameter tuning. The zone MPC resulted in a periodic

operation in the target zone. While both MPC designs succeeded in maintaining

the system operation in the target zone, the zone MPC apparently achieves better

transient economic performance.

Further, we compared the performance of set-point MPC and zone MPC in the

presence of small disturbances. An unknown normally distributed process disturbance

w(n) ∼ N (0, 0.1) is introduced to the system: x(n + 1) = x(n)2 + u(n) + w(n).

89

0 10 20 30 40 50 60 70-1.5

-1-0.5

00.5

x

0 10 20 30 40 50 60 70

n

-1.1

-1

-0.9u

Figure 5.3: State and input trajectories of set-point MPC.

0 10 20 30 40 50 60 70-1.5

-1-0.5

00.5

x

0 10 20 30 40 50 60 70

n

-1.05

-1

-0.95

u

Figure 5.4: State and input trajectories of the zone MPC.

0 10 20 30 40 50 60 70-1.5

-1-0.5

00.5

x

0 10 20 30 40 50 60 70

n

-1.05

-1

-0.95

u

Figure 5.5: State and input trajectories of set-point MPC.

90

0 10 20 30 40 50 60 70-1.5

-1-0.5

00.5

x

0 10 20 30 40 50 60 70

n

-1.05

-1

-0.95u

Figure 5.6: State and input trajectories of the zone MPC.

The state and input trajectories of the set-point MPC and zone MPC are shown

in Figure 5.5 and Figure 5.6, respectively. It is seen that in the presence of small

disturbances, zone MPC is still capable of achieving optimal economic performance

while satisfying the zone objective, whereas set-point MPC inevitably suffers from

fluctuations in the control input.

More insights into the superiority of zone control over set-point control can be

acquired by observing the bifurcation diagram in Figure 5.7, which shows the stable

steady states, periodic orbits and chaotic attractors of system (5.10). We can see that

the target zone Zt contains a set of unstable steady states as well as a set of stable

periodic orbits with period 2. Thus, it is clear how relaxing a set-point target to a

zone target can allow for more admissible operations, thereby releasing more degrees

of freedom to achieve economic objectives.

5.7 Summary

In this work, we proposed a general framework of nonlinear model predictive control

for zone tracking. The target zone is characterized by coupled system state and input,

and is not necessarily control invariant. An invariance-like theorem is developed which

naturally generalizes LaSalle’s invariance principle from autonomous system to control

systems with a zone target. Our results differ from the standard stability analysis for

conventional set-point MPC in that we consider the evolvement of the state-input pair

(x(n), u(n)) instead of merely the the state trajectory x(n). This provides stronger

91

Figure 5.7: Bifurcation diagram of system (5.10). The bifurcation diagram shows the thevalues of system state x(n) that are visited or approached asymptotically under a constantu(n). For u ∈ (−0.75,−0.6], the system has stable steady-state operation. At u = −0.75,steady-state operation becomes unstable and the diagram bifurcates into two points whichcorresponds to a stable periodic operation of period two. As the value of u decreases, morebifurcation occurs and at some point around u = −1.4 the system enters a chaotic operatingregion. The dashed line corresponds to steady-state operating points

results and more accurate description of system behavior. Two simulations examples

demonstrate the superiority of zone control over set-point control and the efficacy of

the proposed zone MPC framework.

92

Chapter 6

Application to an Oilsand PrimarySeparation Vessel

In this chapter, we apply the proposed EMPC algorithm to an oilsand primary sep-

aration vessel (PSV). We show how previously developed EMPC design and analysis

results in the context of discrete-time system can be extended to continuous-time

systems where the issue of sampling needs to be addressed. Both infinite-time and

finite-time operations are considered which correspond to the result in Chapter 2 and

Chapter 4.

6.1 Preliminaries

6.1.1 Notation

Throughout this chapter, the operator | · | denotes the Euclidean norm of a scalar

or a vector. A continuous function α : [0, a) → [0,∞) is said to belong to class K

if it is strictly increasing and satisfies α(0) = 0. The symbol Ωr is used to denote a

level set of the Lyapunov function V (x) such that Ωr := x ∈ Rnx : V (x) ≤ r. The

symbol diag(x1, ..., xi, ..., xn) denotes a diagonal matrix with its i-th diagonal element

being xi. The symbol ‘\’ denotes set substraction such that A \B := x ∈ A, x /∈ B.

The operator d·e is used to obtain the smallest integer greater than or equal to a real

number x ∈ R such that dxe := mina ∈ Z : a ≥ x where Z denotes the set of

integers.

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We consider a class of nonlinear systems which can be described by the following

state-space model:

x(t) = f(x(t)) + g(x(t))u(t) (6.1)

where x ∈ Rnx denotes the state vector and u ∈ Rnu denotes the control input vector.

The control input is restricted to be in a nonempty convex set U ⊂ Rnu , such that:

U := u ∈ Rnu : |u| ≤ umax (6.2)

where umax is the magnitude of the input constraint. It is assumed that f and g are

locally Lipschitz functions with respect to x, and that f(0) = 0 which implies that the

origin is a steady state of the unforced nominal system. We assume that the system

state are sampled periodically at time instants tk≥0 such that tk = t0 + k∆ with

t0 = 0 being the initial time, ∆ being a fixed sampling interval and k being positive

integers.

6.1.3 Stability assumption

We assume that there exists a continuous controller u = h(x) which renders the

origin of system (6.1) asymptotically stable while satisfying the input constraint.

Based on converse Lyapunov theorem [63], the above assumption implies that there

exists a continuous differentiable control Lyapunov function V (x) and K functions

αj, j = 1, 2, 3, 4, that satisfy the following conditions:

α1(|x|) ≤ V (x) ≤ α2(|x|)∂V (x)

∂x

(f(x) + g(x)h(x)

)≤ −α3(|x|)∣∣∣∣∂V (x)

∂x

∣∣∣∣ ≤ α4(|x|)

h(x) ∈ U

(6.3)

for all x ∈ D where D ⊂ Rnx is a compact set containing the origin. We denote a

level set of V (x) in D, Ωρ ⊂ D, as the stability region of the closed-loop system under

the controller h(x).

Using the Lipschitz properties assumed for f , g, and taking into account the

boundedness of the input vector u characterized in Eq. (6.2), there exits a positive

94

constant M such that

|f(x) + g(x)u| ≤M (6.4)

for all x ∈ Ωρ. In addition, by the continuous differentiable property of V (x) and the

Lipschitz properties of f and g, there exist positive constants Lf , Lg, such that:∣∣∣∣∂V (x)∂x

f(x)− ∂V (x′)∂x

f(x′)

∣∣∣∣ ≤ Lf |x− x′|∣∣∣∣∂V (x)∂x

g(x)− ∂V (x′)∂x

g(x′)

∣∣∣∣ ≤ Lg |x− x′|(6.5)

for all x, x′ ∈ Ωρ.

6.2 EMPC with terminal cost

In this section, we present the proposed EMPC with terminal cost and focus on

infinite-time operation. First, we highlight the stability properties of the closed-loop

system under the controller h(x) when it is implemented in a sample-and-hold fashion.

Second, we discuss the design of the terminal cost and then introduce the proposed

EMPC design. Nominal closed-loop stability and asymptotic average performance of

the proposed EMPC design are subsequently analyzed.

6.2.1 Properties of the nonlinear controller h(x)

Since the nonlinear controller h(x) ensures the asymptotic stability of the origin, it

is natural to take advantage of it as the benchmark control system. We are aimed

to develop an EMPC that gives closed-loop economic performance better than (or at

least as good as) the one given by h(x).

In this chapter, we consider that the controller h(x) is implemented in a sample-

and-hold fashion. The sample-and-hold implementation facilitates the construction of

a terminal cost that will be used in the proposed EMPC. When h(x) is implemented

in sample-and-hold, the sampling time ∆ plays an important role in ensuring the

closed-loop stability. If the sampling time is not appropriately picked, the closed-loop

stability under h(x) may be lost. The following proposition summarizes the stability

properties of the controller h(x) implemented in sample-and-hold.

95

Proposition 3 Consider the state trajectory x(t) of the system of Eq. (6.1) in closed-

loop under the controller h(x) implemented in a sample-and-hold fashion with the

initial state x(t0) ∈ Ωρ. If the sampling time ∆ is chosen such that ∆ < ∆max where

∆max ∈ (0,√

ρ(Lf+Lgumax)M

] and is the solution to d(∆max) = 0 with the function d(s)

defined as follows:

d(s) = −α3(α−12 (ρ− (Lf + Lgu

max)Ms2)) + (Lf + Lgumax)Ms (6.6)

with class K functions α2, α3 defined in Eq. (6.3), Lf , Lg defined in Eq. (6.5), and

M defined in Eq. (6.4), then x(t) will be driven into Ωρ∗ ⊂ Ωρ in N∗ steps without

leaving the stability region, where

ρ∗ = α2(α−13 ((Lf + Lgu

max)M∆)) + (Lf + Lgumax)M∆2 (6.7)

and

N∗ =

⌈ρ− ρ∗

−α3(α−12 (ρ∗))∆ + (Lf + Lgumax)M∆2

⌉. (6.8)

Moreover, once x(t) enters Ωρ∗, it is maintained in it thereafter.

Proof. First, we show that ∆max that satisfies Eq. (6.6) uniquely exists. It can

be verified that d(s) is a strictly increasing function for s ∈ [0,√

ρ(Lf+Lgumax)M

]. It

can also be easily checked that d(0) < 0 and d(√

ρ(Lf+Lgumax)M

) > 0. Therefore the

∆max that satisfies d(∆max) = 0 for ∆max ∈ (0,√

ρ(Lf+Lgumax)M

] uniquely exists.

Second, we show that when ∆ ≤ ∆max, the stability conditions established in [64]

are satisfied and the system state will be driven into Ωρ∗ in N∗ steps and is maintained

in it thereafter. Note first that ρ∗ < ρ holds automatically which can be verified from

the definition of ∆max and the fact that ∆ < ∆max. From Eq. (6.7), the following

inequality holds:

−α3(α−12 (ρ∗)) + (Lf + Lgu

max)M∆ < 0 (6.9)

From the analysis in [64], the above equation implies that

V (x(t)) < −α3(α−12 (ρ∗)) + (Lf + Lgu

max)M∆, ∀ x(t) ∈ Ωρ \ Ωρ∗ (6.10)

which further implies that any x(t0) ∈ Ωρ \Ωρ∗ will be driven into Ωρ∗ in less than N∗

steps while V (x) keeps decreasing. To show that once x enters Ωρ∗ it is maintained in

96

it thereafter, we note that for any ρs ∈ (ρo, ρ], where ρo = α2(α−13 ((Lf+Lgu

max)M∆)),

the following inequality holds

−α3(α−12 (ρs)) + (Lf + Lgu

max)M∆ < 0 (6.11)

Using the result in [64] again, the above inequality implies that x is ultimately main-

tained in Ωρ∗o , where ρ∗o := maxV (x(t + ∆)) : V (x(t)) ≤ ρo. Further, for any

x(tk) ∈ Ωρo , the change rate of the Lyapunov function V (x) is bounded as follows for

t ∈ [tk, tk+1]:

V (t) < −α3(α−11 (ρo)) + (Lf + Lgu

max)M∆ < (Lf + Lgumax)M∆ (6.12)

Taking into account the definition of ρ∗o and the above inequality, the following upper

bound on ρ∗o can be obtained:

ρ∗o < ρo + (Lf + Lgumax)M∆2 = ρ∗ (6.13)

which means that x is ultimately maintained in Ωρ∗

It can be seen from Proposition 3 that the nonlinear controller h(x) implemented

in sample-and-hold is capable of maintaining system state inside the stability region

if the sampling time ∆ is sufficiently small. Moreover, it will drive the system state

into a level set Ωρ∗ ⊂ Ωρ in N∗ steps. Note that ρ∗ and N∗ are both monotonic

functions of ∆ for ∆ ∈ (0,∆max). Smaller sampling time ∆ results in smaller ρ∗ and

larger N∗. Moreover, arbitrarily small ρ∗ can be obtained if the sampling time ∆ can

be made arbitrarily small.

Remark 13 Note that the stability property of h(x) implemented in sample-and-hold

in Proposition 3 is made possible due to the continuous differentiability of the Lya-

punov function V (x) as well as the Lipschitz properties of f and g. Note also that the

result in Proposition 3 is conservative in the sense that the actual region in which x is

maintained after N∗ steps might be much smaller than Ωρ∗ characterized by Eq. (6.7)

because worst case is assumed throughout the analysis.

6.2.2 Proposed EMPC design

To proceed, we define x(t|tk) and u(t|tk) as the predicted state and input trajecto-

ries of the system of Eq. (6.1) under the sample-and-hold implementation of h(x)

97

evaluated at tk with the initial condition x(tk|tk) = x(tk). Specifically, for a time

interval t ∈ [tk, tk+Nh ], the predicted state and input trajectories can be obtained by

integrating the following differential equation recursively for l = 0, ..., Nh − 1:

˙x(t|tk) = f(x(t|tk)) + g(x(t|tk))u(t|tk), t ∈ [tk+l, tk+l+1) (6.14a)

u(t|tk) = h(x(tk+l|tk)), t ∈ [tk+l, tk+l+1) (6.14b)

x(tk|tk) = x(tk) (6.14c)

A function, c(x(tk), Nh), that characterizes the accumulated economic performance

of the nominal system under h(x) implemented in sample-and-hold over Nh sampling

periods with the initial state x(tk|tk) = x(tk) can thus be defined as follows:

c(x(tk), Nh) =

∫ tk+Nh

tk

l(x(t|tk), u(t|tk))dt (6.15)

where l(x, u) is a bounded function defined on (x, u) : x ∈ Ωρ, u ∈ U that charac-

terizes the dynamic economic cost of the system at a specific time instant.

In the proposed design, EMPC minimizes the economic cost function l(x, u) for a

prediction horizon of N sampling periods with the cost function c(x,Nh) incorporated

as the terminal cost. The system state is allowed to be operated only in the stability

region Ωρ. Specifically, the proposed EMPC at a sampling time tk is formulated as

the following optimization problem:

minu(τ)∈S(∆)

∫ tk+N

tk

l(x(τ), u(τ))dτ + c(x(tk+N), Nh) (6.16a)

s.t. ˙x(t) = f(x(t)) + g(x(t))u(t) (6.16b)

x(tk) = x(tk) (6.16c)

u(t) ∈ U (6.16d)

x(t) ∈ Ωρ (6.16e)

where S(∆) denotes the family of continuous piece-wise functions with sampling time

∆ and x denotes the predicted state trajectory in this optimization problem.

In the above optimization problem, Eq. (6.16a) is the objective function involving

the accumulated economic performance over the prediction horizon (i.e., the first

term in the objective function) and a terminal cost c(x(tk+N), Nh). Eq. (6.16b) is the

system model. Eq. (6.16c) specifies the initial condition at time tk. Eq. (6.16d) is

98

the constraint on control input u. Eq. (6.16e) specifies the region in which EMPC is

allowed to operate which is used to ensure the closed-loop stability of the proposed

EMPC.

The optimal solution to the optimization problem of Eq. (6.16) is denoted as

u∗(t|tk), t ∈ [tk, tk+N), with the corresponding optimal state trajectory denoted as

x∗(t|tk), t ∈ [tk, tk+N). The manipulated input of the closed-loop system under the

proposed EMPC at a sampling instant tk is defined as follows:

u(t) = u∗(t|tk), t ∈ [tk, tk+1) (6.17)

which is the first value of the optimal input sequence. At the next sampling instant

tk+1, the optimization problem of Eq. (6.16) is re-evaluated.

6.2.3 Stability and performance analysis

In this subsection, we analyze the stability and economic performance of the system

under the proposed EMPC design of Eq. (6.16). The stability property of the proposed

EMPC design is presented in the following theorem.

Theorem 15 Consider the system of Eq. (6.1) in closed-loop under the proposed

EMPC design of Eq. (6.16) with the initial state x(t0) ∈ Ωρ. If the sampling time ∆

satisfies ∆ < ∆max, then the optimization problem is feasible for all tk ≥ t0, and the

system state x(t) ∈ Ωρ for all t ≥ t0.

Proof. We use induction to prove this theorem. At a sampling time tk, let us assume

that x(tk) ∈ Ωρ. If the sampling time ∆ satisfies ∆ < ∆max, it can be obtained from

Proposition 1 that the predicted trajectory x(t|tk) following Eq. (6.15) with Nh = N

is maintained in Ωρ for t ∈ [tk, tk+N) (i.e., constraint (6.16e) is satisfied). Also, the

corresponding u(t|tk) for t ∈ [tk, tk+N) satisfies the input constraint (6.16d). This

implies that the optimization problem of Eq. (6.16) is feasible at tk. Moreover, the

feasibility of the optimization problem of Eq. (6.16) at tk implies that x(t) ∈ Ωρ for

t ∈ [tk, tk+1] since the first step input value is the actual input applied to the system.

x(tk+1) ∈ Ωρ further implies that the optimization problem of Eq. (6.16) at tk+1 is

also feasible. If x(t0) ∈ Ωρ, following the reasoning of induction, the optimization

99

problem of Eq. (6.16) is feasible for all tk ≥ t0, and x(t) ∈ Ωρ for all t ≥ t0. This

proves Theorem 15.

Next, we analyze the closed-loop economic performance of the proposed EMPC. In

particular, we focus on the asymptotic average economic performance since infinite-

time operation is considered. The asymptotic average economic performance is de-

fined as follows:

Jasy := limF→∞

1

F∆

∫ tF

t0

l(x(t), u(t))dt (6.18)

Theorem 2 below provides an upper bound on the asymptotical average performance

of the proposed EMPC design.

Theorem 16 Consider the system of Eq. (6.1) in closed-loop under the proposed

EMPC design of Eq. (6.16) with the initial state x(t0) ∈ Ωρ and sampling time

∆ < ∆max. If Nh is chosen such that Nh ≥ N∗, then the asymptotical average eco-

nomic performance of the closed-loop system under the EMPC, JEMPCasy , is bounded

as follows:

JEMPCasy ≤ Jh∆ (6.19)

where

Jh∆ := max

1

∆

∫ ∆

0

l(x(t), h(x(0)))dt : x(0) ∈ Ωρ∗

(6.20)

which denotes the maximum possible average economic performance over one sampling

period ∆ under the nonlinear controller h(x) implemented in sample-and-hold when

the closed-loop system state is within Ωρ∗.

Proof. Let us use L(tk) to denote the objective function value of the optimization

problem of Eq. (6.16) evaluated at sampling time tk. L(tk) corresponds to the accumu-

lated economic performance of the open-loop system under the following augmented

input sequence implemented in sample-and-hold for t ∈ [tk, tk+N+Nh ]:

U∗(tk) = [u∗(tk|tk), ..., u∗(tk+N−1|tk), h(x(tk+N |tk+N)), ..., h(x(tk+N+Nh−1|tk+N))]

(6.21)

where x(t|tk+N) for t ∈ [tk+N , tk+N+Nh ] denotes the predicted state trajectory under

the nonlinear controller h(x) implemented in sample-and-hold starting from x∗(tk+N |tk).

We will refer to U∗(tk) as the optimal augmented solution to the optimization problem

of Eq. (6.16) at tk.

100

Since x∗(tk+N |tk) ∈ Ωρ (due to Eq. (6.16e)) and Nh ≥ N∗, from Proposition 3,

it can be obtained that the augmented optimal input sequence U∗(tk) will drive the

predicted state trajectory into the invariant set Ωρ∗ , that is:

x(tk+N+Nh |tk+N) ∈ Ωρ∗ (6.22)

Let us further construct the following augmented input sequence:

U(tk+1) = [u∗(tk+1|tk), ..., u∗(tk+N−1|tk), h(x(tk+N |tk+N)), ..., h(x(tk+N+Nh|tk+N))]

(6.23)

which is obtained by discarding the first value in U∗(tk) and applying h(x) for one

more step at the end. It can be shown that U(tk+1) is a feasible augmented solution

to the optimization problem of Eq. (6.16) at tk+1. To verify this, we only need

to check whether h(x(tk+N |tk+N)) satisfies the input constraint of Eq. (6.16d) and

whether the state trajectory x(t|tk+N) for t ∈ [tk+N , tk+N+1] satisfies the constraint

of Eq. (6.16e). Both of these can be verified based on the properties of h(x) and the

fact that x∗(tk+N |tk) ∈ Ωρ.

Let L(tk+1) denote the objective function corresponding to U(tk+1). From the

definition of U(tk+1), L(tk+1) can be expressed as follows:

L(tk+1) = L(tk)−∫ tk+1

tkl(x∗(t|tk), u∗(tk|tk))dt

+∫ tk+N+Nh+1

tk+N+Nh

l(x(t|tk+N), h(x(tk+N+Nh|tk+N))dt(6.24)

Since u∗(tk|tk) is the actually implemented control input, the second term of the right-

hand-side of the above equation can be replaced with the economic performance of

the actual state and input trajectories∫ tk+1

tk

l(x∗(t|tk), u∗(tk|tk))dt =

∫ tk+1

tk

l(x(t), u(t))dt. (6.25)

Moreover, taking into account the definition of Jh∆ and the fact that x(tk+N+Nh|tk+N) ∈

Ωρ∗ , the following inequality holds:∫ tk+N+Nh+1

tk+N+Nh

l(x(t|tk+N), h(x(tk+N+Nh|tk+N))dt ≤ Jh∆∆ (6.26)

Substituting Eq. (6.25) and Eq. (6.26) into Eq. (6.24), the following inequality can

be obtained:

L(tk+1) ≤ L(tk)−∫ tk+1

tk

l(x(t), u(t))dt+ Jh∆∆ (6.27)

101

The principle of optimality indicates that L(tk+1) ≤ L(tk+1), which gives:

L(tk+1) ≤ L(tk)−∫ tk+1

tk

l(x(t), u(t))dt+ Jh∆∆ (6.28)

Summing the above inequality from k = 0 to F − 1 on both sides and combining the

integration terms, the following inequality can be obtained:∫ tF

t0

l(x(τ), u(τ))dτ ≤ L(t0)− L(tF ) + Jh∆F∆ (6.29)

Dividing F∆ on both sides with F approaching infinity, and taking into account

that L(tk) at any specific sampling instant is a finite value because it is a finite time

integration of the bounded cost function l(x, u), the following inequality is obtained:

JEMPCasy = lim

F→∞

1

F∆

∫ tF

t0

l(x(τ), u(τ))dτ ≤ limF→∞

L(t0)− L(tF )

F∆+ Jh∆ = Jh∆ (6.30)

This proves Theorem 16.

Remark 14 In Theorem 16, Jh∆ characterizes the worst possible average economic

performance over one sampling period ∆ that may be obtained when h(x) is imple-

mented in sample-and-hold and the system state is inside the invariant set Ωρ∗. When

the region Ωρ∗ is sufficiently small, if we further assume that the performance metric

l(x, u) is smooth, it is expected that different trajectories in Ωρ∗ will give approximately

the same economic performance. This further implies that:

Jh∆ ≈ limF→∞

1

F∆

∫ F∆

0

l(x, h)dt (6.31)

where x(0) ∈ Ωρ∗. Since h is able to drive the state to Ωρ∗, the above approxima-

tion (6.31) also holds when x(0) ∈ Ωρ. This implies that:

Jh∆ ≈ Jhasy (6.32)

for x(0) ∈ Ωρ. Therefore, the statement in Theorem 1 may also implies that the

asymptotic average performance of the proposed LEMPC is no worse than the asymp-

totic average performance of the controller h.

102

6.3 Fixed-time implementation

In the previous section, a terminal cost EMPC based on the auxiliary controller h(x)

is proposed in which the terminal cost c(x,Nh) is predetermined. It is shown that the

proposed terminal cost EMPC achieves guaranteed asymptotical average performance

over infinite time operation. In this section, we focus on finite-time operation. We

propose an revised implementation strategy of the proposed EMPC with terminal

cost that achieves improved economic performance over the auxiliary controller h(x)

implemented in sample-and-hold for a fixed time duration longer than the prediction

horizon.

6.3.1 Implementation strategy

We consider the performance of the EMPC in a fixed time interval [t0, tF ] where

F > N . The proposed fixed-time EMPC implementation consists of two stages.

In the first stage when tk ∈ [t0, tF−N ], the proposed EMPC design of Eq. (6.16) is

implemented with Nh shrinking every sampling time from F −N to 0; in the second

stage when tk ∈ [tF−N+1, tF ] the proposed EMPC design of Eq. (6.16) is implemented

with Nh = 0 and the prediction horizon shrinking every sampling time from N to 0.

The specific EMPC formulations at the two stages are as follows:

Stage 1 : At a sampling instant tk, tk ∈ [t0, tF−N ], the proposed EMPC design is

formulated as follows:

minu(τ)∈S(∆)

∫ tk+N

tk

l(x(τ), u(τ))dτ + c(x(tk+N), F − k −N) (6.33a)

s.t. ˙x(τ) = f(x(τ)) + g(x(τ))u(τ) (6.33b)

x(tk) = x(tk) (6.33c)

u(τ) ∈ U (6.33d)

x(τ) ∈ Ωρ (6.33e)

The first step value of the optimal solution to the optimization problem of Eq. (6.33) is

implemented between tk and tk+1. At the next sampling instant tk+1, the optimization

problem of Eq. (6.33) is re-evaluated until tk = tF−N .

103

Stage 2 : At a sampling instant tk, tk ∈ [tF−N+1, tF ], the proposed EMPC design

is formulated as follows:

minu(τ)∈S(∆)

∫ tF

tk

l(x(τ), u(τ))dτ (6.34a)

s.t. ˙x(τ) = f(x(τ)) + g(x(τ))u(τ) (6.34b)

x(tk) = x(tk) (6.34c)

u(τ) ∈ U (6.34d)

x(τ) ∈ Ωρ (6.34e)

Again, the first step value of the optimal solution to the optimization problem of Eq.

(6.34) is implemented. At the next sampling instant tk+1, the optimization problem

of Eq. (6.34) is re-evaluated until tk = tF .

6.3.2 Stability and performance analysis

In this subsection, we analyze the stability and economic performance of the proposed

fixed-time EMPC implementation. The stability property of the fixed-time EMPC

implementation is presented in the following theorem.

Theorem 17 Consider the system of Eq. (6.1) in closed-loop under the fixed-time

EMPC implementation of Eq. (6.33) and Eq. (6.34) with the initial state x(t0) ∈ Ωρ.

If the sampling time ∆ satisfies ∆ < ∆max, then the optimization problems are feasible

for all tk ∈ [t0, tF ], and the system state x(t) ∈ Ωρ for all t ∈ [t0, tF ].

Proof. The fixed-time EMPC designs of Eq. (6.33) and Eq. (6.34) are in the same

form of the EMPC of Eq. (6.16). Following similar arguments as in the proof of The-

orem 15, the closed-loop stability of the proposed fixed-time EMPC implementation

can be proved.

The economic performance of the fixed-time EMPC implementation is summarized

in Theorem 18 below.

Theorem 18 Consider the accumulated closed-loop economic performance of the sys-

tem of Eq. (6.1) for a fixed time interval [t0, tF ], with the initial state x(t0) ∈ Ωρ and

sampling time ∆ < ∆max. The accumulated economic performance of the closed-loop

104

system under the fixed-time EMPC implementation of Eq. (6.33) and Eq. (6.34) is

no worse than the accumulated economic performance of the closed-loop system under

h(x) implemented in sample-and-hold; that is:∫ tF

t0

l(x(t), u(t))dt ≤ c(x(t0), F ) (6.35)

where x and u are the actual optimal state and input trajectories under the proposed

fixed-time EMPC implementation strategy.

Proof. Let us use L(tk) to denote the objective function value of the optimization

problem of Eq. (6.33) or Eq. (6.34) evaluated at sampling time tk. L(tk) corresponds

to the accumulated economic performance of the open-loop predicted system state

trajectory under the following input or augmented input sequence implemented in

sample-and-hold for t ∈ [tk, tF ]:

U∗(tk) = [u∗(tk|tk), ..., u∗(tF−1|tk)]. (6.36)

Note that when tk ≤ tF−N , U∗(tk) for the optimization problem of Eq. (6.33) in stage

1 is the optimal augmented input sequence as follows:

U∗(tk) = [u∗(tk|tk), ..., u∗(tk+N−1|tk), h(x(tk+N |tk+N)), ..., h(x(tk+F−1|tk+N))] (6.37)

It can be easily verified that the input sequence U(tk+1) = [u∗(tk+1|tk), ..., u∗(tF−1|tk)]

is a feasible solution to the EMPC optimization problems at tk+1. The objective func-

tion corresponding to this input sequence can be expressed as follows:

L(tk+1) = L(tk)−∫ tk+1

tk

l(x(t), u(t))dt (6.38)

The principle of optimality indicates that L(tk+1) ≤ L(tk+1), which gives:

L(tk+1) ≤ L(tk)−∫ tk+1

tk

l(x(t), u(t))dt (6.39)

Summing the above inequality from k = 0 to F − 1 on both sides and combining the

integration terms, the following inequality can be obtained:∫ tF

t0

l(x(τ), u(τ))dτ ≤ L(t0)− L(tF ) (6.40)

105

At the sampling time tF ,

L(tF ) =

∫ tF

tF

l(x(τ), u(τ))dτ = 0 (6.41)

Because the input sequence generated by h(x) in sample-and-hold from t0 to tF−1 with

initial state x(t0) ∈ ρ is a feasible augmented solution to the optimization problem of

Eq. (6.33) at t0 , the following inequality holds:

L(t0) ≤ c(x(t0), F ) (6.42)

Substituting Eq. (6.41) and Eq. (6.42) into Eq. (6.40):∫ tF

t0

l(x(t), u(t))dt ≤ c(x(t0), F ) (6.43)

This proves Theorem 18.

Remark 15 We note that the shrinking horizon approach as used in Stage 2 may

be used directly starting from t0 with the initial horizon N = F and the results in

Theorem 18 still hold. Similar ideas have been used in [65, 30] to ensure finite-time

performance of EMPC. However, the computational demand of such an implemen-

tation could be very high or even prohibitive for online implementation especially

when F is very large. Taking advantage of the terminal cost, the proposed fixed-time

EMPC implementation is capable of achieving improved fixed-time performance over

the auxiliary controller h(x) with any predetermined prediction horizon N which is

an appealing feature for online implementation.

Remark 16 At the second stage of the proposed fixed-time implementation strategy,

the system state trajectory may be governed by the Turnpike property [66]. In this

case, the system state will deviate from the steady state at the end of the fixed time

window. An alternative implementation strategy to circumvent the dramatic deviation

is to implement the auxiliary controller h(x) instead of the shrinking horizon EMPC

at stage 2. If h(x) is implemented at stage 2, the stability and performance analysis in

Theorem 17 and Theorem 18 still hold. This implementation strategy will be compared

with the strategy outlined in Section 6.3.1 in Section 6.4.

106

6.4 Application to a primary separation vessel

6.4.1 Process description and modeling

The primary separation vessel (PSV) is a key unit in the Clark hot water extraction

process which is used for extraction of all commercial minable oil sands. As the first

step in the extraction process, PSV facilitates the separation of aerated bitumen and

coarse solid particles (clay, sand, etc.). Figure 6.1 shows a schematic of the PSV

process. Oilsand ore is first fed into a mixing tank in which it is mixed with flood

Figure 6.1: Schematic of the primary separation vessel

water, air and other process additives. The mixed oilsand slurry is then fed into

the middle of the primary separation vessel in which gravity separation takes place

due to the density difference of bitumen droplets, water and solids. A bitumen rich

stream (froth) is withdrawn on top of the vessel and sent to downstream units for

further processing. There are also streams withdrawn from the middle and the bottom

layers. The stream withdrawn from the middle layer primarily contains water while

the stream from the bottom layer mostly contains solids.

In the PSV process model, we assume that there are three typical bitumen parti-

cles and three solid particles in the feed stream. The mixer is modeled as a continuous

stirred tank. The separation vessel is modeled into three distinguishable layers: top

froth, middlings and tailings [67]. Each layer is assumed to be well mixed and all com-

position densities are considered constant. The froth/middlings interface level, which

is a crucial control index [68], is dynamically modeled. The middlings/tailings inter-

face is assumed to be stationary (i.e., volume of the tailings layer is constant). Based

107

on mass balance and standard modeling assumptions, a set of ordinary differential

equations is derived to describe the dynamics of the system [67]:

dαfdbjdt

=1

Vmix(Qore(α

orebj − α

fdbj )−Qflα

fdbj ) (6.44a)

dαfdsjdt

=1

Vmix(Qore(α

oresj − α

fdsj )−Qflα

fdsj ) (6.44b)

dVfdt

= vIA (6.44c)

dαfbjdt

=

1Vf

(−αmbjvmbjA+ vIA(αmbj − αfbj)−Qfα

fbj), if vmbj − vI < 0

1Vf

(−αfbjvmbjA−Qfαfbj), if vmbj − vI ≥ 0

(6.44d)

dαfsjdt

=

1Vf

(−αmsjvmsjA+ vIA(αmsj − αfsj)−Qfα

fsj), if vmsj − vI < 0

1Vf

(−αfsjvmsjA−Qfαfsj), if vmsj − vI ≥ 0

(6.44e)

dαmbjdt

=

1Vm

(Qfdαfdbj −Qmα

mbj − αmbjvtbjA+ αmbjv

mbjA), vmbj − vI < 0

1Vm

(Qfdαfdbj −Qmα

mbj − αmbjvtbjA+ αfbjv

mbjA+ vIA(αmbj − α

fbj)), vmbj − vI ≥ 0

(6.44f)

dαmsjdt

=

1Vm

(Qfdαfdsj −Qmα

msj − αmsjvtsjA+ αmsjv

msjA), vmsj − vI < 0

1Vm

((Qfdαfdsj −Qmα

msj − αmsjvtsjA+ αfsjv

msjA+ vIA(αmsj − α

fsj)), vmsj − vI ≥ 0

(6.44g)

dαtbjdt

=1

Vt(αmbjv

tbjA−Qtα

tbj) (6.44h)

dαtsjdt

=1

Vt(αmsjv

tsjA−Qtα

tsj) (6.44i)

where j = 1, 2, 3 stands for the j-th species of bitumen or solids. The definition

of model variables are given in Table 6.1. The values of constant model parameters

are given in Table 6.2. In this model, the movement of all particles is assumed to

be one-dimensional with the downward direction taken as the positive direction. The

volumes of each layers satisfy:

Vm + Vf + Vt − V = 0 (6.45)

The volumetric flows satisfy the following volume balance equations:

Qore +Qfl = Qfd = Qf +Qm +Qt (6.46)

The relative slip velocities of the particles are calculated using the following hindered

108

settling model [69] :

vibj =gd2

bjF (αiw)(ρbj − ρi)18µw

(6.47a)

visj =gd2

sjF (αiw)(ρsj − ρi)18µw

(6.47b)

where i = f, m, t, stands for the froth layer, middlings layer and tailings layer,

respectively. The function F (αiw) is the Barnea and Mizrachi correlation [70]:

F (αiw) = [1 + (1− αiw)13 e

5(1−αiw)

3αiw ]−1 (6.48)

The volumetric fraction of water in layer i, i = f, m, t, satisfies the overall volume

balance in layer i:

αiw = 1−3∑j=1

αibj −3∑j=1

αisj (6.49)

The suspension in layer i, i = f, m, t, is assumed to be uniform and its density is

calculated as the weighted summation of component densities:

ρi = ρwαiw +

3∑j=1

ρbjαibj +

3∑j=1

ρsjαisj (6.50)

The froth/middlings interface velocity, vI , is modeled by first order approximation of

the shockwave equation [71]:

vI =

3∑j=1

αmbjvmbj −

3∑j=1

αfbjvfbj

3∑j=1

αmbj −3∑j=1

αfbj

(6.51)

6.4.2 EMPC design

The system state vector which includes the volume of froth layer and the volumet-

ric concentrations of all bitumen and solid species of the feed stream, froth layer,

middlings layer and tailings layer, is defined as follows:

x = [ Vf αfdb1 αfdb2 αfdb3 αfds1 αfds2 αfds3 αfb1 αfb2 αfb3 αfs1 αfs2 αfs3αmb1 αmb2 αmb3 αms1 αms2 αms3 αtb1 αtb2 αtb3 αts1 αts2 αts3 ]T

(6.52)

109

Table 6.1: Process variables.

Qore, Qfl, Qfd, Qf , Qm, Qt volumetric flow rates of the oilsand ore, flood water,feed stream, froth stream, middlings stream andtailings stream

V, Vf , Vm, Vt, Vmix volumes of the vessel, froth layer, middlings layer,tailings layer and the feed flow mixer

A cross section area of the vesselvI froth/middlings interface velocity

αfdbj , αfbj, α

mbj , α

tbj volumetric fractions of the j-th bitumen in feed

stream, froth layer, middlings layer and tailingslayer

αfdsj , αfsj, α

msj, α

tsj volumetric fractions of the j-th solid in feed stream,

froth layer, middlings layer and tailings layerαfdw , α

fw, α

mw , α

tw volumetric fractions of water in feed stream, froth

layer, middlings layer and tailings layerαorebj , α

oresj volumetric fractions of the j-th bitumen and sands

in the ore

vfbj, vmbj , v

tbj slip velocities of the j-th bitumen in froth layer,

middlings layer and tailings layer

vfsj, vmsj , v

tsj slip velocities of the j-th solid in froth layer, mid-

dlings layer and tailings layerρbj, ρsj, ρw the j-th bitumen particle, j-th solid particle and

water densitiesdbj, dsj the j-th bitumen particle and solid particle diame-

tersρf , ρm, ρt suspension densities of froth layer, middlings layer

and tailings layerµw viscosity of waterg gravitation costant

110

Table 6.2: Model parameters.

ρb = [800, 750, 700] kg/m3 A = 8.5 m2

ρs = 2650 kg/m3 µw = 3.25× 10−4 kg/(m · s)ρw = 971.8 kg/m3 g = 9.81m/s2

αoreb = [0.021, 0.038, 0.051] db = [2.8, 5.8, 7.0]× 10−5 mαores = [0.0168, 0.042, 0.630] ds = [0.78, 1.311, 12.5]× 10−5 mV = 1000 m3 Vt = 200 m3

Vmix = 4.15 m3 Qore = 0.0056m3/s

Table 6.3: Steady-state operating point.

αfdb = [0.0075, 0.0136, 0.0183] αfds = [0.0603, 0.0151, 0.2262]

αfb = [0.0648, 0.2019, 0.2918] αfs = [0, 0, 0]αmb = [0.0232, 0.0159, 0.0149] αms = [0.2982, 0.0598, 0.0315]αtb = [0, 0, 0] αts = [0.0106, 0.0060, 0.2861]Qfl = 0.0100 m3/s Qf = 0.0008m3/sQm = 0.0027 m3/s Qt = 0.0120m3/sVf = 200.0 m3

The input vector which includes flow rates of flood water, middlings stream and

tailings stream, is defined as follows:

u = [u1 u2 u3]T = [Qfl Qm Qt]T (6.53)

The economic objective of the PSV process is to maximize the overall recovery rate

of bitumen from feed ore, which is equivalent to minimizing the time integration of

the following function:

r(x(t), u(t)) = −∑3

j=1 αfbj(t)Qf (t)∑3

j=1 αorebj Qore

= −(x8(t) + x9(t) + x10(t)

)(Qore + u1(t)− u2(t)− u3(t)

)∑3j=1 α

orebj Qore

(6.54)

111

The optimal steady state operating point is obtained by solving the following steady-

state optimization problem:

(xs, us) = arg min r(x, u)

s.t. f(x) = 0

u ∈ U(6.55)

where f(x) is the right-hand-side of the process model of Eq. (6.44) and the control

input is bounded inside the convex set U defined as follows:

U :=

(u1, u2, u3) :

0 ≤ u1 ≤ 0.01

0 ≤ u2 ≤ 0.03

0 ≤ u3 ≤ 0.03

Qf = Qore + u1 − u2 − u3 ≥ 0

(6.56)

The obtained optimal steady state is shown in Table 6.3. The economic cost function

of EMPC is defined as follows:

l(x, u) = r(x, u) + d(u) (6.57)

where the term d(u(t)) penalizes the incremental of the inputs and is defined as

follows:

d(u(t)) = |u(t)− u(tk−1)|2R, t ∈ (tk−1, tk] (6.58)

where R = diag(104, 104, 104). The incremental penalty term d(u(t)) is incorporated

into the economic cost function to make the optimal solution unique while not chang-

ing the optimal steady state. The nonlinear auxiliary controller h(x) is chosen as a

single proportional controller paired between the froth volume Vf and the middling

outlet flow rate Qm, specifically:

h(x) = [u1s, u2s + p(x1 − x1s), u3s]T (6.59)

where the proportional gain is p = −10−7. A quadratic Lyapunov function:

V (x) = (x− xs)TP (x− xs) (6.60)

with P = diag(10−4, 1, 1, ..., 1) is used. A ρ = 0.2 level set of the quadratic Lyapunov

function is chosen as the stability region in which the system state is allowed to

evolve. The corresponding operating region for the froth layer volume is 155.3m3 <

112

Table 6.4: Transient and steady-state average bitumen recovery rates of the closed-loopsystem under the proposed EMPC with terminal cost, the EMPC in [1], the tracking MPCand the controller h(x)

Proposed EMPC EMPC in [1] Tracking MPC h(x)Transient average 0.8816 0.8123 0.7754 0.7690

Steady state 0.7514 0.7389 0.7611 0.7611

Vf < 244.7m3. The sampling time is ∆ = 1 h, the prediction horizon is N = 5, and

the terminal cost evaluation step is Nh = 30.

In addition, the proposed EMPC design will be compared with the conventional

tracking MPC. The stage cost of the tracking MPC is lt(x, u) = |x− xs|2Q + |u− us|2Rwhere the weighting matrices areQ = diag(10−4, 1, 1, ..., 1) andR = diag(104, 104, 104).

6.4.3 Simulation result

In this subsection, the performance of the proposed EMPC design will be compared

with 1) an EMPC without terminal cost [1], 2) the conventional tracking MPC and

3) the controller h(x). The initial state of all simulation runs is:

x(0) = [190.0 0.0516 0.2106 0.3225 0 0 0 0.0168 0.0151

0.0151 0.1865 0.0459 0.2179 0 0 0 0.0020 0.0014

0.6097 0.0122 0.0220 0.0296 0.0975 0.0244 0.3656 ]T

(6.61)

We first compare the performance of the proposed EMPC with terminal cost

for infinite-time operation with the other control configurations. The closed-loop

bitumen recovery rate trajectories, r(x(t), u(t)), under different controllers are shown

in Fig. 6.2. The closed-loop froth layer volume trajectories are shown in Fig. 6.3.

The closed-loop trajectories of the quadratic Lyapunov function of Eq. (6.60), which

indicate the overall state evolutions, are shown in Fig. 6.4. The closed-loop input

trajectories are shown in Fig. 6.5. The transient average recover rate for t ∈ [0, 72h]

and the steady state recovery rates achieved by different controllers are shown in

Table 6.4.

From these results, it can be seen that the controller h(x) and the tracking MPC

eventually drive the system state to the optimal steady state. The two EMPC config-

113

0 20 40 60 720

2

4

t (h)

r

Figure 6.2: Bitumen recovery rate trajectores of the closed-loop system under the proposedEMPC with terminal cost (dashed line), the EMPC in [1] (dotted line), the tracking MPC(dash dotted line) and the controller h(x) (solid line).

0 20 40 60 72165

175

185

195

205

210

t (h)

Vf (

m3 )

Figure 6.3: Froth layer volume trajectores of the closed-loop system under the proposedEMPC with terminal cost (dashed line), the EMPC in [1] (dotted line), the tracking MPC(dash dotted line) and the controller h(x) (solid line).

114

0 20 40 60 720

0.05

0.1

0.15

0.2

t (h)

V(x

)

Figure 6.4: Lyapunov function trajectores of the closed-loop system under the proposedEMPC with terminal cost (dashed line), the EMPC in [1] (dotted line), the tracking MPC(dash dotted line) and the controller h(x) (solid line).

0 10 20 30 40 50 60 700

2

4x 10

−3

Qm

(m

3 /s)

0 10 20 30 40 50 60 700.005

0.01

0.015

Qt (

m3 /s

)

t (h)

0 10 20 30 40 50 60 706

8

10x 10

−3

Qfl (

m3 /s

)

Figure 6.5: Input trajectories of the closed-loop system under the proposed EMPC withterminal cost (dashed line), the EMPC in [1] (dotted line), the tracking MPC (dash dottedline) and the controller h(x) (solid line).

115

Table 6.5: Average bitumen recovery rates of the closed-loop system under (a): theinfinite-time implementation, (b): fixed-time implementation, (c): alternative fixed-timeimplementation, and (d): the controller h(x).

(a) (b) (c) (d)Average recovery rate 0.8816 1.0191 0.8798 0.7690

urations (with and without terminal cost) drive the system state to new steady states

slightly different from the optimal steady state while maintaining the system state

inside the stability region. Note that this result agrees with the performance analysis

in Theorem 2. It can be seen from Fig. 6.4 that in Nh = 30 steps, the controller

h(x) drives the system state to a small region around the optimal steady state with

ρ∗ = 0.004.

Note that the EMPCs achieve higher transient average recovery rates than the

conventional tracking MPC or the controller h(x) because recovery rate is directly

accounted for in the EMPC cost functions. Note also that the proposed EMPC design

achieves both higher transient recovery rate and higher steady-state recovery rate over

the EMPC in [1] because of the enforcement of the proposed terminal cost.

Next, the performance of the fixed-time implementation strategy proposed in Sec-

tion 4 is investigated. The fixed time interval is t ∈ [0, 72h]. Comparisons are made

between the proposed EMPC with terminal cost for infinite-time operation, the pro-

posed fixed-time implementation strategy, the alternative fixed-time implementation

described in Remark 16, and the controller h(x). The closed-loop bitumen recovery

rate trajectories under different controllers and implementation strategies are shown

in Fig. 6.6. The closed-loop froth layer volume trajectories are shown in Fig. 6.7. The

closed-loop trajectories of the quadratic Lyapunov function are shown in Fig. 6.8. The

closed-loop input trajectories are shown in Fig. 6.9. The average recovery rates and

steady-state recovery rates achieved by different implementations for the fixed 72h

time interval are shown in Table 6.5.

From these results, it can be seen that fixed-time implementation strategies achieve

improved average bitumen recovery rates over the controller h(x) for the fixed time

interval, which agrees with the analysis of Theorem 4. It can be also seen that the

116

0 20 40 60 720

2

4

6

8

t (h)

r

Figure 6.6: Bitumen recovery rates of the closed-loop system under the infinite-timeimplementation (dashed line), fixed-time implementation(dotted line), alternative fixed-time implementation (dash dotted line) and the controller h(x) (solid line).

0 20 40 60 72165

175

185

195

205

t (h)

Vf (

m3 )

Figure 6.7: Froth layer volume trajectories of the closed-loop system under the infinite-time implementation (dashed line), fixed-time implementation(dotted line), alternativefixed-time implementation (dash dotted line) and the controller h(x) (solid line).

117

0 20 40 60 720

0.05

0.1

0.15

0.2

t (h)

V(x

)

Figure 6.8: Lyapunov function trajectories of the closed-loop system under the infinite-time implementation (dashed line), fixed-time implementation(dotted line), alternativefixed-time implementation (dash dotted line) and the controller h(x) (solid line).

0 10 20 30 40 50 60 700

5x 10

−3

Qm

(m

3 /s)

0 10 20 30 40 50 60 700

0.01

0.02

t (h)

Qt (

m3 /s

)

0 10 20 30 40 50 60 700

0.005

0.01

Qfl (

m3 /s

)

Figure 6.9: Input trajectories of the closed-loop system under the infinite-time implemen-tation (dashed line), fixed-time implementation(dotted line), alternative fixed-time imple-mentation (dash dotted line) and the controller h(x) (solid line).

118

fixed-time implementation strategy proposed in Section 4.1 leads to highest average

recovery rate for the fixed time interval with its system trajectory exhibiting the

turnpike property. The dramatic deviation at the end is undesirable and is fixed by

the alternative implementation described in Remark 16. In the first stage interval

t ∈ [0, 67h], the closed-loop system evolvements under the proposed EMPC design

for infinite-time and finite-time operations remain basically the same until a slight

divergence occurs at the end when Nh for the finite-time operation gets smaller. This

result indicates that the proposed EMPC design is insensitive to the value of Nh

when it is sufficiently large, which agrees with the properties of the controller h(x).

We would also like to note that it is not established that the fixed-time EMPC gives

better performance than the infinite-time EMPC or vice versa. The performance

depends on many different factors such as the duration of fix-time operation, the

EMPC prediction horizon and the the auxiliary controller.

6.5 Summary

In this chapter, a terminal cost construction approach was developed for EMPC and

EMPC algorithms were designed for both infinite-time and finite-time operations.

In the proposed approach, an auxiliary nonlinear controller that renders the desired

optimal steady state asymptotically stable was taken advantage of. The two EMPC

algorithms give provable improved economic performance than the auxiliary controller

with very flexible requirement on the length of the prediction horizon. This means

that the proposed EMPC algorithms could be very computationally efficient. The

proposed EMPC algorithms were applied to an oilsand primary separation vessel.

The simulation results demonstrated the effectiveness of the proposed approaches.

119

Chapter 7

Conclusions and Future Work

7.1 Conclusions

In this thesis, we systematically discussed the stability and performance of the general

EMPC scheme with extended horizon, and explored its extension or application to

several specific scenarios.

We proposed the basic EMPC formulation with extended prediction horizon based

on an auxiliary controller In Chapter 2. In the analysis, special attention was paid to

the impact of the terminal horizon on the stability and performance of the proposed

EMPC design. While a finite terminal horizon is in general not sufficient to ensure

stability of the optimal steady state, it is sufficient to achieve practical stability for

strictly dissipative systems under mild assumptions. Further conditions to ensure the

exponential shrinkage of the practical stability region are provided. For a special case

including conventional MPC with positive-definite stage costs, exponential stability

can be achieved. Performance of the EMPC is also shown to be approximately upper-

bounded by that of the auxiliary controller if a large terminal horizon is used. These

results provide insights into the intrinsic properties of EMPC and also explain the

computational efficiency of the EMPC design.

Chapters 3-6 are all extensions of the proposed EMPC design from Chapter 2. In

Chapter 3, we design a terminal cost for economic model predictive control (EMPC)

which preserves local optimality. We first showed, based on the strong duality and

second order sufficient condition (SOSC) of the steady-state optimization problem,

that the optimal operation of the system is locally equivalent to an infinite-horizon

LQR controller. The proposed terminal cost is constructed as the value function

120

of the LQR controller plus a linear term characterized by the Lagrange multiplier

associated with the steady state constraint. EMPC with the proposed terminal cost

is stabilizing with an appropriately chosen control horizon, and preserves the local

optimality of the LQR controller.

In Chapter 4, we extended the proposed EMPC design to control systems with

scheduled switching operations. The proposed EMPC scheme takes advantage of a set

of auxiliary controllers that locally stabilizes the optimal steady state of each operat-

ing mode. In the proposed approach, EMPC operations are divided into two phases

— an infinite-time operation phase and a mode transition phase, depending on the

current sampling time and the scheduled mode switching time. Sufficient conditions

to ensure recursive feasibility of the proposed EMPC design are established. The

proposed EMPC design is computationally efficient and enjoys enlarged feasibility

regions than the auxiliary controllers. Simulation results of a chemical process exam-

ple demonstrate the superiority of our design over existing MPC designs for switched

scheduling operations.

In Chapter 5, we proposed a general framework of nonlinear model predictive

control for zone tracking. The target zone is characterized by coupled system state

and input, and is not necessarily control invariant. An invariance-like theorem is

developed which naturally generalizes LaSalle’s invariance principle from autonomous

system to control systems with a zone target. Our results differ from the standard

stability analysis for conventional set-point MPC in that we consider the evolvement

of the state-input pair (x(n), u(n)) instead of merely the the state trajectory x(n).

This provides stronger results and more accurate description of system behavior.

Two simulations examples demonstrate the superiority of zone control over set-point

control and the efficacy of the proposed zone MPC framework.

In Chapter 6, we applied the proposed EMPC algorithm to an oilsand primary

separation vessel (PSV). We showed how previously developed EMPC design and

analysis results in the context of discrete-time system can be extended to continuous-

time systems where the issue of sampling were addressed.

121

7.2 Future research directions

Distributed EMPC. Distributed MPC has been a heated research area in recent years

and is widely regarded as the most effective way to address organizational and com-

putational issues of large-scale systems. Since EMPC involves general economic cost

functions that are nonlinear and possibly non-convex, the computational complexity

of EMPC is in general higher than conventional set-point tracking MPC. This makes

the motivation for a distributed control framework even stronger for EMPC. Some

existing results on distributed EMPC include [72, 73, 74, 75, 76]. It would be interest-

ing to investigate EMPC with extended horizon in a distributed control framework.

For example, based on the topology of the system and the communication network,

a centralized auxiliary controller or a set of decentralized auxiliary controllers can be

designed offine. In the online optimization, each EMPC unit will utilize the auxil-

iary controllers to predict the long term effects of their individual behavior on the

overall networked control system. This could ensure the stability and asymptotic

performance of the networked control system. Fundamental research on networked

dissipative systems will be needed which requires knowledge from game theory, topol-

ogy and graph theory. Practical issues encountered in the communication network

such as asynchronous measurements and measurement delays and their influences on

economic performance also deserve future research attention.

EMPC with optimal non-steady-state operation. In this thesis we have mainly

focused on strictly dissipative systems whose infinite-time optimal operations are

steady-state operation. However, for the general case, non-steady-state optimal op-

erations are possible. On the one hand, generic nonlinear systems may feature non-

steady optimal equilibrium solution such as optimal periodic solution. On the other

hand, time-varying economic cost or disturbances may render the optimal operation

non-steady. Recently, there are pioneering works on extending dissipativity analysis

to periodic MPC design [77]. However, existing results rely on prior knowledge of

the optimal steady state or the optimal periodic orbit. In the case of handling time-

varying objectives and operating conditions, set-point independent EMPC design and

analysis are needed.

Robust EMPC. Robustness is yet another untouched area in this thesis. It is

122

conceivable that EMPC lack robustness compared to the conventional MPC, because

conventional MPC is designed to stabilize the process while EMPC is designed to

pursue economic performance. Earlier studies on the inherent robustness of MPC

relied on the continuity of the value function [54] [32]. Recent development of MPC

robustness analysis has been aided with the theory of input-to-state stability (ISS)

[60], which offers an elegant way of proving robustness by finding an ISS-Lyapunov

function. It will be rather straightforward to establish ISS stability of the proposed

EMPC with extended horizon if the auxiliary controller is continuous and the terminal

constraint set is inactive. Since the extended horizon is expected to add a certain

degree of robustness to EMPC, it would also be interesting to investigate the extent

to which extended horizon adds to the EMPC robustness.

123

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